Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential...

18
Research Article Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application Kai Du 1 and Zhen Wu 2 1 School of Mathematics and Zhongtai Securities Institute for Financial Study, Shandong University, Jinan 250100, China 2 School of Mathematics, Shandong University, Jinan 250100, China Correspondence should be addressed to Zhen Wu; [email protected] Received 5 October 2018; Accepted 5 February 2019; Published 21 February 2019 Academic Editor: Konstantina Skouri Copyright © 2019 Kai Du and Zhen Wu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. en the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high- dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained. 1. Introduction e stochastic differential games are important in various fields such as biology, engineering, economics, management, and particularly financial investment, and they are useful in modeling dynamic systems involving noise terms where more than one decision maker are involved. Among various differential games, the Stackelberg game, a concept of a hierarchical solution for markets where some firms have power of domination over others, is firstly introduced by H. von Stackelberg in 1934 [1]. Since then, a lot of literature is studied to deal with the deterministic Stackelberg game, such as Basar and Olsder [2], Long [3]. For the stochas- tic cases, Bagchi and Basar [4] studied an LQ stochastic Stackelberg differential game, where the state and control variables do not enter the diffusion coefficient in the state equation. Yong [5] obtained a more general result, with random coefficients, control dependent diffusion, and the weight matrices for the controls in the cost functionals being not necessarily positive definite. Bensoussan et al. [6] investigated a stochastic Stackelberg differential game in various information structures (i.e., adapted open-loop), whereas the diffusion coefficient does not contain the con- trol variables. Furthermore, for the open-loop information structure and closed-loop memoryless information structure cases, they give the corresponding two types of optimal strategies of forward stochastic Stackelberg differential game. In Øksendal et al. [7], a time-dependent newsvendor problem with time-delayed information is solved, based on stochastic Stackelberg differential game approach. Shi et el. [8] solved a stochastic Stackelberg differential game with asymmet- ric information. Since the theory of mean-field forward stochastic differential equations (MF-SDEs) studied by [9], their related topics and applications (particularly in financial engineering) have been investigated by many authors (see [10–13]). Based on above, [14] studied the open-loop LQ Stackelberg game of the mean-field stochastic systems in finite horizon and got the feedback representation of the open-loop Stackelberg equilibrium involving the new state and its mean. Here, we point out that all references mentioned above focus on the Stackelberg game with forward state equation in which the initial condition is specified at initial time. However, in financial investment, one frequently encoun- ters financial investment problems with future conditions (as random variables at terminal time ) specified. us, by contrast, this paper introduces a stochastic Stackelberg differential game following a linear MF-BSDE. e general Hindawi Mathematical Problems in Engineering Volume 2019, Article ID 1798585, 17 pages https://doi.org/10.1155/2019/1798585

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Research ArticleLinear-Quadratic Stackelberg Game for Mean-Field BackwardStochastic Differential System and Application

Kai Du 1 and ZhenWu 2

1School of Mathematics and Zhongtai Securities Institute for Financial Study Shandong University Jinan 250100 China2School of Mathematics Shandong University Jinan 250100 China

Correspondence should be addressed to ZhenWu wuzhensdueducn

Received 5 October 2018 Accepted 5 February 2019 Published 21 February 2019

Academic Editor Konstantina Skouri

Copyright copy 2019 KaiDu andZhenWuThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations(MF-BSDEs) By means of four Riccati equations (REs) the follower first solves a backward mean-field stochastic LQ optimalcontrol problem and gets the corresponding open-loop optimal control with the feedback representation Then the leader turnsto solve an optimization problem for a 1 times 2 mean-field forward-backward stochastic differential system In virtue of some high-dimensional and complicated REs we obtain the open-loop Stackelberg equilibrium and it admits a state feedback representationFinally as applications a class of stochastic pension fund optimization problems which can be viewed as a special case of ourformulation is studied and the open-loop Stackelberg strategy is obtained

1 Introduction

The stochastic differential games are important in variousfields such as biology engineering economics managementand particularly financial investment and they are usefulin modeling dynamic systems involving noise terms wheremore than one decision maker are involved Among variousdifferential games the Stackelberg game a concept of ahierarchical solution for markets where some firms havepower of domination over others is firstly introduced by Hvon Stackelberg in 1934 [1] Since then a lot of literatureis studied to deal with the deterministic Stackelberg gamesuch as Basar and Olsder [2] Long [3] For the stochas-tic cases Bagchi and Basar [4] studied an LQ stochasticStackelberg differential game where the state and controlvariables do not enter the diffusion coefficient in the stateequation Yong [5] obtained a more general result withrandom coefficients control dependent diffusion and theweight matrices for the controls in the cost functionalsbeing not necessarily positive definite Bensoussan et al[6] investigated a stochastic Stackelberg differential gamein various information structures (ie adapted open-loop)whereas the diffusion coefficient does not contain the con-trol variables Furthermore for the open-loop information

structure and closed-loop memoryless information structurecases they give the corresponding two types of optimalstrategies of forward stochastic Stackelberg differential gameInOslashksendal et al [7] a time-dependent newsvendor problemwith time-delayed information is solved based on stochasticStackelberg differential game approach Shi et el [8] solveda stochastic Stackelberg differential game with asymmet-ric information Since the theory of mean-field forwardstochastic differential equations (MF-SDEs) studied by [9]their related topics and applications (particularly in financialengineering) have been investigated by many authors (see[10ndash13]) Based on above [14] studied the open-loop LQStackelberg game of the mean-field stochastic systems infinite horizon and got the feedback representation of theopen-loop Stackelberg equilibrium involving the new stateand its mean

Here we point out that all references mentioned abovefocus on the Stackelberg game with forward state equationin which the initial condition is specified at initial timeHowever in financial investment one frequently encoun-ters financial investment problems with future conditions(as random variables at terminal time 119879) specified Thusby contrast this paper introduces a stochastic Stackelbergdifferential game following a linear MF-BSDE The general

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1798585 17 pageshttpsdoiorg10115520191798585

2 Mathematical Problems in Engineering

backward stochastic differential equations (BSDEs) wereinitially studied by [15 16] and extended to mean-field caseintroduced by [17]The BSDEs are well-formulated stochasticsystems and have found various applications El Karoui etal [18] gave some important properties of BSDEs and theirapplications to optimal controls and financial mathematicsKohlmann and Zhou [19] studied the relationship betweena BSDE and a forward LQ optimal control problem andbased on it Lim and Zhou [20] discussed a backward LQoptimal control problem Li et al [21] studied the backwardLQ optimal control problem for mean-field case Huang etal [22] studied a backward LQ optimal control in partialinformation and gave some applications in pension fundoptimization problems Furthermore some recent literaturecan be found in [23ndash26] for the study of games followingbackward stochastic differential systems and [21] for thestudy of control problems following mean-filed backwardstochastic differential systems

Inspired by above-mentioned motivations this paperstudies a new kind of Stackelberg differential game for mean-field backward stochastic differential system Specifically weconsider stochastic dynamic games involving two leaderand follower agents satisfying linear MF-BSDE systems Itdistinguishes itself from the literaturementioned above in thefollowing aspects

(i) An important class of Stackelberg differential gameof MF-BSDEs is introduced which consists of twostochastic optimal control problems (ie a stochasticoptimal control problem of MF-BSDEs for the fol-lower and a stochastic optimal control problem ofmean-field forward-backward stochastic differentialequations (MF-FBSDEs) for the leader) Unlike for-ward MF-SDEs the solution of MF-BSDEs shouldconsist of one adapted solution pair (119910(sdot) 119911(sdot)) (see(1)) where the second component 119911(sdot) is naturallyintroduced to ensure the adaptiveness of 119910(sdot) whenpropagating from terminal backward to initial time

(ii) For the follower the open-loop optimal control of hisLQ control problem is characterized in terms of theMF-FBSDE (10)-(11) To get the state feedback repre-sentations for the optimal controls of the follower weintroduce some new equations four Riccati equationsa mean-field stochastic differential equation and amean-field backward stochastic differential equation

(iii) For the leaderrsquos optimal control problem of mean-field forward-backward stochastic differential sys-tem under standard conditions we conclude theuniqueness and existence of an optimal control fromwhich the cost functional is strictly convex andcoercive with respect to the control variable Byvirtue of the maximum principle method the open-loop optimal control can be represented via theHamiltonian system and adjoint process Moreoverstate feedback representation for the optimal controlsof the leader is explicitly given with the help ofsome new high-dimensional and coupled Riccatiequations

(iv) Last but not least we study a class of stochasticpension fund optimization problem with two rep-resentative members Applying the aforementionedconclusions we derive the open-loop optimal contri-bution policy in feedback representation

Some remarks to above points are given as follows Asto (ii) unlike the standard Hamiltonian system for MF-SDEs control which is a MF-FBSDE coupled in its terminalcondition the Hamiltonian system in BSDE control setupbecomes a mean-field forward-backward stochastic differen-tial equation (MF-FBSDE) coupled in its initial conditionTherefore to decouple it and get the feedback representationwe should introduce some additional REs and MF-SDEs Asto (iii) since more equations are introduced to decouple theHamiltonian system of the follower the dynamic process ofthe leader becomes a ldquo1 times 2rdquo MF-FBSDE (one forward andtwo backward mean-field stochastic differential equations)which is different from those of forward case (the stateprocess for forward case is an ldquo1 times 1rdquo MF-FBSDE)

The structure of this paper is as follows Section 1 givesthe introduction and specifies some standard notations andterminologies Section 2 formulates the LQ Stackelberg gamefor MF-BSDEs The corresponding optimal control problemof MF-BSDEs for the follower is studied in Section 3Section 4 studies the stochastic optimal control problem ofMF-FBSDEs for the leader and the open-loop Stackelbergstrategy in feedback representation is obtainedThe stochasticpension fund optimization problems with two representativemembers are studied in Section 5 Section 6 concludes ourwork and presents some future research direction

11 Notation and Terminology The following notations willbe used throughout the paper We let R119899 be the Euclideanspace of 119899-dimensional Euclidean spaceR119899times119889 be the space of119899times119889matrices andS119899 be the space of 119899times119899 symmetricmatrices⟨sdot sdot⟩ and | sdot | denote the scalar product and norm in theEuclidean space respectively The transpose of a vector (ormatrix) 119909 is denoted by 119909⊤ If119872 isin S119899 is positive semidefinitewe write 119872 ge 0 Consider a finite time horizon [0 119879] fora fixed 119879 gt 0 Let 119883 be a given Hilbert space The set of119883-valued continuous functions is denoted by 119862([0 119879]119883) If119873(sdot) isin 119862([0 119879] 119878119899) and 119873(119905) ge 0 for every 119905 isin [0 119879] wesay that 119873(sdot) is positive semidefinite which is denoted by119873(sdot) ge 0

We suppose (ΩF F P) is a complete filtered probabil-ity space on which a standard one-dimensional Brownianmotion119882 = 119882(119905)0le119905le119879 is defined where F = F119905119905ge0 is thenatural filtration of119882 augmented by all the P-null sets inFSuppose 120589 Ω 997888rarr R119899 is an F119879-random variable We write120589 isin 1198712F119879(ΩR119899) if 120589 is square integrable (ie E|120589|2 lt +infin)Consider 119891 [0 119879] times Ω 997888rarr R119899 is a F adapted processIf 119891(sdot) is square integrable (ie E int119879

0|119891(119905)|2119889119905 lt infin ) we

shall write 119891(sdot) isin 1198712F (0 119879R119899) if 119891(sdot) is uniformly bounded(ie esssup(119905120596)isin[0119879]timesΩ|119891(119905)| lt infin) then 119891(sdot) isin 119871infinF (0 119879R119899)These definitions generalize in the obvious way to the casewhen 119891(sdot) is R119899times119898 (or S119899) valued Furthermore in caseswherewe restrict ourselves to deterministic Borelmeasurablefunctions 119891 [0 119879] 997888rarr R119899 we shall drop the subscript F

Mathematical Problems in Engineering 3

in the notation for example 119871infin(0 119879R119899) Finally we denote1199092119877 ≜ ⟨119877119909 119909⟩ for all 119877 isin S119899 and 119909 isin R119899

2 Problem Formulation

In this paper we consider the following controlled linear MF-BSDE119889119910 (119905) = minus [119860 (119905) 119910 (119905) + 119860 (119905)E119910 (119905) + 1198611 (119905) 1199061 (119905)+ 1198611 (119905)E1199061 (119905) + 1198612 (119905) 1199062 (119905) + 1198612 (119905)E1199062 (119905)+ 119862 (119905) 119911 (119905) + 119862 (119905)E119911 (119905)] 119889119905 + 119911 (119905) 119889119882 (119905) 119910 (119879) = 120585

(1)

Here (119910(sdot) 119911(sdot)) isin R119899 times R119899 is the state process Note that119911(sdot) is also part of solution of (1) which is introduced here toensure the adaptiveness of 119910(sdot) 1199061(sdot) isin U1[0 119879] is the controlprocess of the follower 1199062(sdot) isin U2[0 119879] is the control processof the leader and the admissible control sets are given by

U119894 [0 119879] ≜ 1198712F (0 119879R119898119894) 119894 = 1 2 (2)

respectively 119860(sdot) 119860(sdot) 1198611(sdot) 1198611(sdot) 1198612(sdot) 1198612(sdot) 119862(sdot) 119862(sdot) aregiven deterministic matrix-valued functions 120585 isin 1198712F119879(ΩR119899)is the terminal condition Now we introduce the followingassumption that will be in force throughout this paper

(H1) The coefficients of the state equation satisfy thefollowing119860 (sdot) 119860 (sdot) 119862 (sdot) 119862 (sdot) isin 119871infin (0 119879R119899times119899) 1198611 (sdot) 1198611 (sdot) isin 119871infin (0 119879R119899times1198981) 1198612 (sdot) 1198612 (sdot) isin 119871infin (0 119879R119899times1198982)

(3)

Under (H1) for all 119906119894 isin U119894[0 119879] 119894 = 1 2 MF-BSDE (1) hasa unique adapted solution belonging to 1198712F (0 119879R119899) (see [21Theorem 21]) Furthermore we define the cost functionals oftwo players as

119869119894 (120585 1199061 1199062) ≜ Eint1198790[⟨119876119894 (119905) 119910 (119905) 119910 (119905)⟩

+ ⟨119876119894 (119905)E119910 (119905) E119910 (119905)⟩ + ⟨119877119894 (119905) 119906119894 (119905) 119906119894 (119905)⟩+ ⟨119877119894 (119905)E119906119894 (119905) E119906119894 (119905)⟩ + ⟨119878119894 (119905) 119911 (119905) 119911 (119905)⟩+ ⟨119878119894 (119905)E119911 (119905) E119911 (119905)⟩] 119889119905 + ⟨119866119894119910 (0) 119910 (0)⟩+ ⟨119866119894E119910 (0) E119910 (0)⟩

(4)

For the coefficients of cost functionals we shall assume thefollowing assumptions throughout this paper

(H2) For all 119894 = 1 2 the weighting coefficients in the costfunctional satisfy119876119894 (sdot) 119876119894 (sdot) 119878119894 (sdot) 119878119894 (sdot) isin 119871infin (0 119879S119899)

119877119894 (sdot) 119877119894 (sdot) isin 119871infin (0 119879S119898119894) 119866119894 119866119894 isin S119899

(5)

and there exists a constant 120575 gt 0 such that for ae 119905 isin [0 119879]119876119894 (119905) 119876119894 (119905) + 119876119894 (119905) 119878119894 (119905) 119878119894 (119905) + 119878119894 (119905) ge 0119877119894 (119905) 119877119894 (119905) + 119877119894 (119905) ge 120575119868119866119894 119866119894 + 119866119894 ge 0

(6)

For the sake of notation simplicity the time argument issuppressed in cost functional above and in the sequel of thispaper wherever necessary

Let us now explain the mean-filed backward stochasticStackelberg differential game system In the game Player 1is the follower and Player 2 is the leader In addition forany 119894 = 1 2 we assume that 119869119894(120585 1199061 1199062) is a cost functionalfor Player 119894 Therefore at the initial time for any giventerminal target 120585 isin 1198712F119879(ΩR119899) the leader announces hisstrategy 1199062 isin U2[0 119879] over the whole planning horizon[0 119879] Then with the knowledge of the leaderrsquos strategy thefollower determines his response strategy 1199061(sdot) isin U1[0 119879]over the entire horizon to minimize 1198691(120585 1199061 1199062) Since thefollowerrsquos optimal response can be determined by the leaderthe leader can take it into account in finding and announcingher optimal strategy which minimizes 1198692(120585 1199061 1199062) over 1199062 isinU2[0 119879] In a little more rigorous way we give the followingdefinition of the Stackelberg game

Definition 1 The pair (1199061 1199062) isin U1[0 119879] timesU2[0 119879] is calledan open-loop solution to the above Stackelberg game if itsatisfies the following conditions

(i) There exists a map Γ U2[0 119879] times 1198712F119879(ΩR119899) 997888rarrU1[0 119879] such that1198691 (120585 Γ (1199062 120585) 1199062) = min

1199061isinU1[0119879]1198691 (120585 1199061 1199062)

forall1199062 isin U2 [0 119879] (7)

(ii) There exists a unique 1199062 isin U2[0 119879] such that1198692 (120585 Γ (1199062 120585) 1199062) = min1199062isinU2[0119879]

1198692 (120585 Γ (1199062 120585) 1199062) (8)

(iii) The optimal strategy of the follower is 1199061 = Γ(1199062 120585)The aim of the paper is to find the feedback represen-

tation of the open-loop Stackelberg strategy for mean-fieldbackward differential game (1) and (4)

3 Optimization for the Follower

In this section we consider the optimization problem of thefollower For given 1199062 isin U2[0 119879] the follower wants to solvethe following LQ optimal control problems for mean-fieldbackward stochastic differential system

4 Mathematical Problems in Engineering

Problem (BMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199061 isinU1[0 119879] such that

1198691 (120585 1199061 1199062) = min1199061isinU1[0119879]

1198691 (120585 1199061 1199062) (9)

By using the similar method found in [21] we are able toobtain the following result

Proposition 2 Under (H1)-(H2) let the terminal state 120585 isin1198712F119905(ΩR119899) and the leaderrsquos strategy 1199062 isin U2[0 119879] be given(i) The problem (BMF-LQ) is uniquely solvable with(119910 119911 1199061) being the only optimal pair if and only if there exists

a unique 4-tuple (119909 119910 119911 1199061) satisfying the MF-FBSDE

119889119909 = (119860⊤119909 + 119860⊤E119909 + 1198761119910 + 1198761E119910)119889119905 + (119862⊤119909+ 119862⊤E119909 + 1198781119911 + 1198781E119911) 119889119882(119905)

119889119910 = minus (119860119910 + 119860E119910 + 11986111199061 + 1198611E1199061 + 11986121199062 + 1198612E1199062+ 119862119911 + 119862E119911) 119889119905 + 119911119889119882 (119905) 119909 (0) = 1198661119910 (0) + 1198661E119910 (0) 119910 (119879) = 120585

(10)

and the following stationarity condition holds

11987711199061 + 1198771E1199061 + 119861⊤1 119909 + 119861⊤1 E119909 = 0 (11)

(2) The following relations are satisfied

E119910 = minus1198752E119909 minus E120601119910 minus E119910 = minus1198751 (119909 minus E119909) minus (120601 minus E120601) E119911 = minus [119868 + 1198751 (1198781 + 1198781)]minus1 [1198751 (119862 + 119862)⊤ E119909 + E120578] 119911 minus E119911 = minus (119868 + 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578]

(12)

where 1198751 1198752 and (120601 120578) are the solutions of the following threeequations respectively

1 + 1198601198751 + 1198751119860⊤ minus 119875111987611198751 + 1198611119877minus11 119861⊤1 + 119862 (119868+ 11987511198781)minus1 1198751119862⊤ = 01198751 (119879) = 0(13)

2 + (119860 + 119860)1198752 + 1198752 (119860 + 119860)⊤ + (1198611 + 1198611) (1198771+ 1198771)minus1 (1198611 + 1198611)⊤ + (119862 + 119862) [119868 + 1198751 (1198781 + 1198781)]minus1sdot 1198751 (119862 + 119862)⊤ minus 1198752 (1198761 + 1198761) 1198752 = 01198752 (119879) = 0(14)

119889120601 = minus [(119860 minus 11987511198761) (120601 minus E120601)+ (119860 + 119860 minus 1198752 (1198761 + 1198761))E120601 minus 11986121199062 minus 1198612E1199062+ 119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 E120578] 119889119905+ 120578119889119882 (119905) 120601 (119879) = minus120585

(15)

In above proposition we get the optimal strategy

1199061 = minus119877minus11 119861⊤1 (119909 minus E119909) minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ E119909 (16)

of the follower with any given leaderrsquos strategy 1199062 isin U2[0 119879]through the adjoint equation 119909 (one part of the Hamiltoniansystem (10)-(11)) Next we intend to obtain the state feedbackform of 1199061 We first have the following lemma

Lemma 3 Under (H1)-(H2) let (119909 119910 119911) be the solution of theHamiltonian system (10)-(11) Then

119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593 (17)

where 1198753 and 1198754 satisfy the following two Riccati equationsrespectively

3 minus 119860⊤1198753 minus 1198761 minus 1198753119860 + 11987531198611119877minus11 119861⊤1 1198753+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤1198753 = 01198753 (0) = 119866(18)

4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤) 1198754 minus (1198761 + 1198761) = 01198754 (0) = 1198661 + G1(19)

and 120593 is given by the following MF-SDE

119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578 119889119882(119905) 120593 (0) = 0

(20)

Mathematical Problems in Engineering 5

where 119860119910 ≜ 119860 + 1198611199101198753119860119910 ≜ 119860 + 119860 + 1198611199101198754119861119910 ≜ minus1198611119877minus11 119861⊤1 119861119910 ≜ minus (119861 + 1198611) (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ 119860120601 ≜ 119860 minus 11987511198761119860120601 ≜ 119860 + 119860 minus 1198752 (1198761 + 1198761) 119862120601 ≜ 119862 (119868 + 11987511198781)minus1 119862120601 ≜ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 119860120593 ≜ [119860119910 minus 1198621206011198751119862⊤1198753]⊤ 119860120593 ≜ [119860119910 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987531198751)minus1 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987541198752)minus1 119863120593 ≜ (1198781 minus 1198753) (119868 + 11987511198781)minus1 119863120593 ≜ (1198781 + 1198781 minus 1198753) (119868 + 1198751 (1198781 + 1198781))minus1

(21)

Proof Since (119909 119910 119911) is the solution of the Hamiltoniansystem (10)-(11) by noticing relations (12) we can get that theinitial value of 119909 satisfies

E119909 (0) = minus (1198661 + 1198661) 1198752 (0)E119909 (0)minus (1198661 + 1198661)E120601 (0) 119909 (0) minus E119909 (0) = minus11986611198751 (0) (119909 (0) minus E119909 (0))minus 1198661 (120601 (0) minus E120601 (0)) (22)

By using the fact that(119868 + 119872119873)minus1119872 = 119872(119868 + 119873119872)minus1 forall119873 119872 isin S119899 (23)

where we assume that 119868 + 119873119872 and 119868 + 119872119873 are reversible itcan be shown by a straightforward computation that119909 (0)= minus1198661 (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))

minus (1198661 + 1198661) [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (24)

which implies119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (25)

Therefore noting (11) and (21) we can rewrite the backwardstochastic differential equation 119910 of FBDSE (10) as thefollowing form MF-SDE

119889119910 = minus 119860119910 + 119860E119910 + (119861119910 minus 1198621206011198751119862⊤) (119909 minus E119909)+ (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909 minus 119862120601 (120578 minus E120578)minus 119862120601E120578 + 11986121199062 + 1198612E1199062 119889119905 minus (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ E119909 + E120578] 119889119882 (119905)

119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0)) minus [119868+ 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0)

(26)

Then we conjecture that 119909 and 119910 are related by the following119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593E119909 = 1198754E119910 + E120593119909 minus 119864119909 = 1198753 (119910 minus E119910) + 120593 minus E120593 (27)

where 1198753 1198754 [0 119879] 997888rarr S119899 are absolutely continuous withinitial value 1198661 1198661 + 1198661 respectively and 120593 satisfies119889120593 = 120572119889119905 + 120573119889119882 (119905) 120593 (0) = 0 (28)

for some F119905-progressively measurable processes 120572 and 120573Note that E119909 and E119910 satisfy the following two ordinarydifferential equations respectively

119889E119909 = [(119860 + 119860)⊤ E119909 + (1198761 + 1198761)E119910] 119889119905119909 (0) = (1198661 + 1198661)E119910 (0) 119889E119910 = minus (119860 + 119860)E119910 + (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909minus 119862120601E120578 + (1198612 + 1198612)E1199062 119889119905

E119910 (0) = minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (29)

Applying Itorsquos formula to the second equation of (27) we get

0 = [4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)1198754 minus (1198761 + 1198761)]E119910+ E120572 minus [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593+ 1198754119862120601E120578 minus 1198754 (1198612 + 1198612)E1199062

(30)

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 2: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

2 Mathematical Problems in Engineering

backward stochastic differential equations (BSDEs) wereinitially studied by [15 16] and extended to mean-field caseintroduced by [17]The BSDEs are well-formulated stochasticsystems and have found various applications El Karoui etal [18] gave some important properties of BSDEs and theirapplications to optimal controls and financial mathematicsKohlmann and Zhou [19] studied the relationship betweena BSDE and a forward LQ optimal control problem andbased on it Lim and Zhou [20] discussed a backward LQoptimal control problem Li et al [21] studied the backwardLQ optimal control problem for mean-field case Huang etal [22] studied a backward LQ optimal control in partialinformation and gave some applications in pension fundoptimization problems Furthermore some recent literaturecan be found in [23ndash26] for the study of games followingbackward stochastic differential systems and [21] for thestudy of control problems following mean-filed backwardstochastic differential systems

Inspired by above-mentioned motivations this paperstudies a new kind of Stackelberg differential game for mean-field backward stochastic differential system Specifically weconsider stochastic dynamic games involving two leaderand follower agents satisfying linear MF-BSDE systems Itdistinguishes itself from the literaturementioned above in thefollowing aspects

(i) An important class of Stackelberg differential gameof MF-BSDEs is introduced which consists of twostochastic optimal control problems (ie a stochasticoptimal control problem of MF-BSDEs for the fol-lower and a stochastic optimal control problem ofmean-field forward-backward stochastic differentialequations (MF-FBSDEs) for the leader) Unlike for-ward MF-SDEs the solution of MF-BSDEs shouldconsist of one adapted solution pair (119910(sdot) 119911(sdot)) (see(1)) where the second component 119911(sdot) is naturallyintroduced to ensure the adaptiveness of 119910(sdot) whenpropagating from terminal backward to initial time

(ii) For the follower the open-loop optimal control of hisLQ control problem is characterized in terms of theMF-FBSDE (10)-(11) To get the state feedback repre-sentations for the optimal controls of the follower weintroduce some new equations four Riccati equationsa mean-field stochastic differential equation and amean-field backward stochastic differential equation

(iii) For the leaderrsquos optimal control problem of mean-field forward-backward stochastic differential sys-tem under standard conditions we conclude theuniqueness and existence of an optimal control fromwhich the cost functional is strictly convex andcoercive with respect to the control variable Byvirtue of the maximum principle method the open-loop optimal control can be represented via theHamiltonian system and adjoint process Moreoverstate feedback representation for the optimal controlsof the leader is explicitly given with the help ofsome new high-dimensional and coupled Riccatiequations

(iv) Last but not least we study a class of stochasticpension fund optimization problem with two rep-resentative members Applying the aforementionedconclusions we derive the open-loop optimal contri-bution policy in feedback representation

Some remarks to above points are given as follows Asto (ii) unlike the standard Hamiltonian system for MF-SDEs control which is a MF-FBSDE coupled in its terminalcondition the Hamiltonian system in BSDE control setupbecomes a mean-field forward-backward stochastic differen-tial equation (MF-FBSDE) coupled in its initial conditionTherefore to decouple it and get the feedback representationwe should introduce some additional REs and MF-SDEs Asto (iii) since more equations are introduced to decouple theHamiltonian system of the follower the dynamic process ofthe leader becomes a ldquo1 times 2rdquo MF-FBSDE (one forward andtwo backward mean-field stochastic differential equations)which is different from those of forward case (the stateprocess for forward case is an ldquo1 times 1rdquo MF-FBSDE)

The structure of this paper is as follows Section 1 givesthe introduction and specifies some standard notations andterminologies Section 2 formulates the LQ Stackelberg gamefor MF-BSDEs The corresponding optimal control problemof MF-BSDEs for the follower is studied in Section 3Section 4 studies the stochastic optimal control problem ofMF-FBSDEs for the leader and the open-loop Stackelbergstrategy in feedback representation is obtainedThe stochasticpension fund optimization problems with two representativemembers are studied in Section 5 Section 6 concludes ourwork and presents some future research direction

11 Notation and Terminology The following notations willbe used throughout the paper We let R119899 be the Euclideanspace of 119899-dimensional Euclidean spaceR119899times119889 be the space of119899times119889matrices andS119899 be the space of 119899times119899 symmetricmatrices⟨sdot sdot⟩ and | sdot | denote the scalar product and norm in theEuclidean space respectively The transpose of a vector (ormatrix) 119909 is denoted by 119909⊤ If119872 isin S119899 is positive semidefinitewe write 119872 ge 0 Consider a finite time horizon [0 119879] fora fixed 119879 gt 0 Let 119883 be a given Hilbert space The set of119883-valued continuous functions is denoted by 119862([0 119879]119883) If119873(sdot) isin 119862([0 119879] 119878119899) and 119873(119905) ge 0 for every 119905 isin [0 119879] wesay that 119873(sdot) is positive semidefinite which is denoted by119873(sdot) ge 0

We suppose (ΩF F P) is a complete filtered probabil-ity space on which a standard one-dimensional Brownianmotion119882 = 119882(119905)0le119905le119879 is defined where F = F119905119905ge0 is thenatural filtration of119882 augmented by all the P-null sets inFSuppose 120589 Ω 997888rarr R119899 is an F119879-random variable We write120589 isin 1198712F119879(ΩR119899) if 120589 is square integrable (ie E|120589|2 lt +infin)Consider 119891 [0 119879] times Ω 997888rarr R119899 is a F adapted processIf 119891(sdot) is square integrable (ie E int119879

0|119891(119905)|2119889119905 lt infin ) we

shall write 119891(sdot) isin 1198712F (0 119879R119899) if 119891(sdot) is uniformly bounded(ie esssup(119905120596)isin[0119879]timesΩ|119891(119905)| lt infin) then 119891(sdot) isin 119871infinF (0 119879R119899)These definitions generalize in the obvious way to the casewhen 119891(sdot) is R119899times119898 (or S119899) valued Furthermore in caseswherewe restrict ourselves to deterministic Borelmeasurablefunctions 119891 [0 119879] 997888rarr R119899 we shall drop the subscript F

Mathematical Problems in Engineering 3

in the notation for example 119871infin(0 119879R119899) Finally we denote1199092119877 ≜ ⟨119877119909 119909⟩ for all 119877 isin S119899 and 119909 isin R119899

2 Problem Formulation

In this paper we consider the following controlled linear MF-BSDE119889119910 (119905) = minus [119860 (119905) 119910 (119905) + 119860 (119905)E119910 (119905) + 1198611 (119905) 1199061 (119905)+ 1198611 (119905)E1199061 (119905) + 1198612 (119905) 1199062 (119905) + 1198612 (119905)E1199062 (119905)+ 119862 (119905) 119911 (119905) + 119862 (119905)E119911 (119905)] 119889119905 + 119911 (119905) 119889119882 (119905) 119910 (119879) = 120585

(1)

Here (119910(sdot) 119911(sdot)) isin R119899 times R119899 is the state process Note that119911(sdot) is also part of solution of (1) which is introduced here toensure the adaptiveness of 119910(sdot) 1199061(sdot) isin U1[0 119879] is the controlprocess of the follower 1199062(sdot) isin U2[0 119879] is the control processof the leader and the admissible control sets are given by

U119894 [0 119879] ≜ 1198712F (0 119879R119898119894) 119894 = 1 2 (2)

respectively 119860(sdot) 119860(sdot) 1198611(sdot) 1198611(sdot) 1198612(sdot) 1198612(sdot) 119862(sdot) 119862(sdot) aregiven deterministic matrix-valued functions 120585 isin 1198712F119879(ΩR119899)is the terminal condition Now we introduce the followingassumption that will be in force throughout this paper

(H1) The coefficients of the state equation satisfy thefollowing119860 (sdot) 119860 (sdot) 119862 (sdot) 119862 (sdot) isin 119871infin (0 119879R119899times119899) 1198611 (sdot) 1198611 (sdot) isin 119871infin (0 119879R119899times1198981) 1198612 (sdot) 1198612 (sdot) isin 119871infin (0 119879R119899times1198982)

(3)

Under (H1) for all 119906119894 isin U119894[0 119879] 119894 = 1 2 MF-BSDE (1) hasa unique adapted solution belonging to 1198712F (0 119879R119899) (see [21Theorem 21]) Furthermore we define the cost functionals oftwo players as

119869119894 (120585 1199061 1199062) ≜ Eint1198790[⟨119876119894 (119905) 119910 (119905) 119910 (119905)⟩

+ ⟨119876119894 (119905)E119910 (119905) E119910 (119905)⟩ + ⟨119877119894 (119905) 119906119894 (119905) 119906119894 (119905)⟩+ ⟨119877119894 (119905)E119906119894 (119905) E119906119894 (119905)⟩ + ⟨119878119894 (119905) 119911 (119905) 119911 (119905)⟩+ ⟨119878119894 (119905)E119911 (119905) E119911 (119905)⟩] 119889119905 + ⟨119866119894119910 (0) 119910 (0)⟩+ ⟨119866119894E119910 (0) E119910 (0)⟩

(4)

For the coefficients of cost functionals we shall assume thefollowing assumptions throughout this paper

(H2) For all 119894 = 1 2 the weighting coefficients in the costfunctional satisfy119876119894 (sdot) 119876119894 (sdot) 119878119894 (sdot) 119878119894 (sdot) isin 119871infin (0 119879S119899)

119877119894 (sdot) 119877119894 (sdot) isin 119871infin (0 119879S119898119894) 119866119894 119866119894 isin S119899

(5)

and there exists a constant 120575 gt 0 such that for ae 119905 isin [0 119879]119876119894 (119905) 119876119894 (119905) + 119876119894 (119905) 119878119894 (119905) 119878119894 (119905) + 119878119894 (119905) ge 0119877119894 (119905) 119877119894 (119905) + 119877119894 (119905) ge 120575119868119866119894 119866119894 + 119866119894 ge 0

(6)

For the sake of notation simplicity the time argument issuppressed in cost functional above and in the sequel of thispaper wherever necessary

Let us now explain the mean-filed backward stochasticStackelberg differential game system In the game Player 1is the follower and Player 2 is the leader In addition forany 119894 = 1 2 we assume that 119869119894(120585 1199061 1199062) is a cost functionalfor Player 119894 Therefore at the initial time for any giventerminal target 120585 isin 1198712F119879(ΩR119899) the leader announces hisstrategy 1199062 isin U2[0 119879] over the whole planning horizon[0 119879] Then with the knowledge of the leaderrsquos strategy thefollower determines his response strategy 1199061(sdot) isin U1[0 119879]over the entire horizon to minimize 1198691(120585 1199061 1199062) Since thefollowerrsquos optimal response can be determined by the leaderthe leader can take it into account in finding and announcingher optimal strategy which minimizes 1198692(120585 1199061 1199062) over 1199062 isinU2[0 119879] In a little more rigorous way we give the followingdefinition of the Stackelberg game

Definition 1 The pair (1199061 1199062) isin U1[0 119879] timesU2[0 119879] is calledan open-loop solution to the above Stackelberg game if itsatisfies the following conditions

(i) There exists a map Γ U2[0 119879] times 1198712F119879(ΩR119899) 997888rarrU1[0 119879] such that1198691 (120585 Γ (1199062 120585) 1199062) = min

1199061isinU1[0119879]1198691 (120585 1199061 1199062)

forall1199062 isin U2 [0 119879] (7)

(ii) There exists a unique 1199062 isin U2[0 119879] such that1198692 (120585 Γ (1199062 120585) 1199062) = min1199062isinU2[0119879]

1198692 (120585 Γ (1199062 120585) 1199062) (8)

(iii) The optimal strategy of the follower is 1199061 = Γ(1199062 120585)The aim of the paper is to find the feedback represen-

tation of the open-loop Stackelberg strategy for mean-fieldbackward differential game (1) and (4)

3 Optimization for the Follower

In this section we consider the optimization problem of thefollower For given 1199062 isin U2[0 119879] the follower wants to solvethe following LQ optimal control problems for mean-fieldbackward stochastic differential system

4 Mathematical Problems in Engineering

Problem (BMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199061 isinU1[0 119879] such that

1198691 (120585 1199061 1199062) = min1199061isinU1[0119879]

1198691 (120585 1199061 1199062) (9)

By using the similar method found in [21] we are able toobtain the following result

Proposition 2 Under (H1)-(H2) let the terminal state 120585 isin1198712F119905(ΩR119899) and the leaderrsquos strategy 1199062 isin U2[0 119879] be given(i) The problem (BMF-LQ) is uniquely solvable with(119910 119911 1199061) being the only optimal pair if and only if there exists

a unique 4-tuple (119909 119910 119911 1199061) satisfying the MF-FBSDE

119889119909 = (119860⊤119909 + 119860⊤E119909 + 1198761119910 + 1198761E119910)119889119905 + (119862⊤119909+ 119862⊤E119909 + 1198781119911 + 1198781E119911) 119889119882(119905)

119889119910 = minus (119860119910 + 119860E119910 + 11986111199061 + 1198611E1199061 + 11986121199062 + 1198612E1199062+ 119862119911 + 119862E119911) 119889119905 + 119911119889119882 (119905) 119909 (0) = 1198661119910 (0) + 1198661E119910 (0) 119910 (119879) = 120585

(10)

and the following stationarity condition holds

11987711199061 + 1198771E1199061 + 119861⊤1 119909 + 119861⊤1 E119909 = 0 (11)

(2) The following relations are satisfied

E119910 = minus1198752E119909 minus E120601119910 minus E119910 = minus1198751 (119909 minus E119909) minus (120601 minus E120601) E119911 = minus [119868 + 1198751 (1198781 + 1198781)]minus1 [1198751 (119862 + 119862)⊤ E119909 + E120578] 119911 minus E119911 = minus (119868 + 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578]

(12)

where 1198751 1198752 and (120601 120578) are the solutions of the following threeequations respectively

1 + 1198601198751 + 1198751119860⊤ minus 119875111987611198751 + 1198611119877minus11 119861⊤1 + 119862 (119868+ 11987511198781)minus1 1198751119862⊤ = 01198751 (119879) = 0(13)

2 + (119860 + 119860)1198752 + 1198752 (119860 + 119860)⊤ + (1198611 + 1198611) (1198771+ 1198771)minus1 (1198611 + 1198611)⊤ + (119862 + 119862) [119868 + 1198751 (1198781 + 1198781)]minus1sdot 1198751 (119862 + 119862)⊤ minus 1198752 (1198761 + 1198761) 1198752 = 01198752 (119879) = 0(14)

119889120601 = minus [(119860 minus 11987511198761) (120601 minus E120601)+ (119860 + 119860 minus 1198752 (1198761 + 1198761))E120601 minus 11986121199062 minus 1198612E1199062+ 119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 E120578] 119889119905+ 120578119889119882 (119905) 120601 (119879) = minus120585

(15)

In above proposition we get the optimal strategy

1199061 = minus119877minus11 119861⊤1 (119909 minus E119909) minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ E119909 (16)

of the follower with any given leaderrsquos strategy 1199062 isin U2[0 119879]through the adjoint equation 119909 (one part of the Hamiltoniansystem (10)-(11)) Next we intend to obtain the state feedbackform of 1199061 We first have the following lemma

Lemma 3 Under (H1)-(H2) let (119909 119910 119911) be the solution of theHamiltonian system (10)-(11) Then

119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593 (17)

where 1198753 and 1198754 satisfy the following two Riccati equationsrespectively

3 minus 119860⊤1198753 minus 1198761 minus 1198753119860 + 11987531198611119877minus11 119861⊤1 1198753+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤1198753 = 01198753 (0) = 119866(18)

4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤) 1198754 minus (1198761 + 1198761) = 01198754 (0) = 1198661 + G1(19)

and 120593 is given by the following MF-SDE

119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578 119889119882(119905) 120593 (0) = 0

(20)

Mathematical Problems in Engineering 5

where 119860119910 ≜ 119860 + 1198611199101198753119860119910 ≜ 119860 + 119860 + 1198611199101198754119861119910 ≜ minus1198611119877minus11 119861⊤1 119861119910 ≜ minus (119861 + 1198611) (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ 119860120601 ≜ 119860 minus 11987511198761119860120601 ≜ 119860 + 119860 minus 1198752 (1198761 + 1198761) 119862120601 ≜ 119862 (119868 + 11987511198781)minus1 119862120601 ≜ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 119860120593 ≜ [119860119910 minus 1198621206011198751119862⊤1198753]⊤ 119860120593 ≜ [119860119910 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987531198751)minus1 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987541198752)minus1 119863120593 ≜ (1198781 minus 1198753) (119868 + 11987511198781)minus1 119863120593 ≜ (1198781 + 1198781 minus 1198753) (119868 + 1198751 (1198781 + 1198781))minus1

(21)

Proof Since (119909 119910 119911) is the solution of the Hamiltoniansystem (10)-(11) by noticing relations (12) we can get that theinitial value of 119909 satisfies

E119909 (0) = minus (1198661 + 1198661) 1198752 (0)E119909 (0)minus (1198661 + 1198661)E120601 (0) 119909 (0) minus E119909 (0) = minus11986611198751 (0) (119909 (0) minus E119909 (0))minus 1198661 (120601 (0) minus E120601 (0)) (22)

By using the fact that(119868 + 119872119873)minus1119872 = 119872(119868 + 119873119872)minus1 forall119873 119872 isin S119899 (23)

where we assume that 119868 + 119873119872 and 119868 + 119872119873 are reversible itcan be shown by a straightforward computation that119909 (0)= minus1198661 (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))

minus (1198661 + 1198661) [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (24)

which implies119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (25)

Therefore noting (11) and (21) we can rewrite the backwardstochastic differential equation 119910 of FBDSE (10) as thefollowing form MF-SDE

119889119910 = minus 119860119910 + 119860E119910 + (119861119910 minus 1198621206011198751119862⊤) (119909 minus E119909)+ (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909 minus 119862120601 (120578 minus E120578)minus 119862120601E120578 + 11986121199062 + 1198612E1199062 119889119905 minus (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ E119909 + E120578] 119889119882 (119905)

119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0)) minus [119868+ 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0)

(26)

Then we conjecture that 119909 and 119910 are related by the following119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593E119909 = 1198754E119910 + E120593119909 minus 119864119909 = 1198753 (119910 minus E119910) + 120593 minus E120593 (27)

where 1198753 1198754 [0 119879] 997888rarr S119899 are absolutely continuous withinitial value 1198661 1198661 + 1198661 respectively and 120593 satisfies119889120593 = 120572119889119905 + 120573119889119882 (119905) 120593 (0) = 0 (28)

for some F119905-progressively measurable processes 120572 and 120573Note that E119909 and E119910 satisfy the following two ordinarydifferential equations respectively

119889E119909 = [(119860 + 119860)⊤ E119909 + (1198761 + 1198761)E119910] 119889119905119909 (0) = (1198661 + 1198661)E119910 (0) 119889E119910 = minus (119860 + 119860)E119910 + (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909minus 119862120601E120578 + (1198612 + 1198612)E1199062 119889119905

E119910 (0) = minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (29)

Applying Itorsquos formula to the second equation of (27) we get

0 = [4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)1198754 minus (1198761 + 1198761)]E119910+ E120572 minus [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593+ 1198754119862120601E120578 minus 1198754 (1198612 + 1198612)E1199062

(30)

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 3: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 3

in the notation for example 119871infin(0 119879R119899) Finally we denote1199092119877 ≜ ⟨119877119909 119909⟩ for all 119877 isin S119899 and 119909 isin R119899

2 Problem Formulation

In this paper we consider the following controlled linear MF-BSDE119889119910 (119905) = minus [119860 (119905) 119910 (119905) + 119860 (119905)E119910 (119905) + 1198611 (119905) 1199061 (119905)+ 1198611 (119905)E1199061 (119905) + 1198612 (119905) 1199062 (119905) + 1198612 (119905)E1199062 (119905)+ 119862 (119905) 119911 (119905) + 119862 (119905)E119911 (119905)] 119889119905 + 119911 (119905) 119889119882 (119905) 119910 (119879) = 120585

(1)

Here (119910(sdot) 119911(sdot)) isin R119899 times R119899 is the state process Note that119911(sdot) is also part of solution of (1) which is introduced here toensure the adaptiveness of 119910(sdot) 1199061(sdot) isin U1[0 119879] is the controlprocess of the follower 1199062(sdot) isin U2[0 119879] is the control processof the leader and the admissible control sets are given by

U119894 [0 119879] ≜ 1198712F (0 119879R119898119894) 119894 = 1 2 (2)

respectively 119860(sdot) 119860(sdot) 1198611(sdot) 1198611(sdot) 1198612(sdot) 1198612(sdot) 119862(sdot) 119862(sdot) aregiven deterministic matrix-valued functions 120585 isin 1198712F119879(ΩR119899)is the terminal condition Now we introduce the followingassumption that will be in force throughout this paper

(H1) The coefficients of the state equation satisfy thefollowing119860 (sdot) 119860 (sdot) 119862 (sdot) 119862 (sdot) isin 119871infin (0 119879R119899times119899) 1198611 (sdot) 1198611 (sdot) isin 119871infin (0 119879R119899times1198981) 1198612 (sdot) 1198612 (sdot) isin 119871infin (0 119879R119899times1198982)

(3)

Under (H1) for all 119906119894 isin U119894[0 119879] 119894 = 1 2 MF-BSDE (1) hasa unique adapted solution belonging to 1198712F (0 119879R119899) (see [21Theorem 21]) Furthermore we define the cost functionals oftwo players as

119869119894 (120585 1199061 1199062) ≜ Eint1198790[⟨119876119894 (119905) 119910 (119905) 119910 (119905)⟩

+ ⟨119876119894 (119905)E119910 (119905) E119910 (119905)⟩ + ⟨119877119894 (119905) 119906119894 (119905) 119906119894 (119905)⟩+ ⟨119877119894 (119905)E119906119894 (119905) E119906119894 (119905)⟩ + ⟨119878119894 (119905) 119911 (119905) 119911 (119905)⟩+ ⟨119878119894 (119905)E119911 (119905) E119911 (119905)⟩] 119889119905 + ⟨119866119894119910 (0) 119910 (0)⟩+ ⟨119866119894E119910 (0) E119910 (0)⟩

(4)

For the coefficients of cost functionals we shall assume thefollowing assumptions throughout this paper

(H2) For all 119894 = 1 2 the weighting coefficients in the costfunctional satisfy119876119894 (sdot) 119876119894 (sdot) 119878119894 (sdot) 119878119894 (sdot) isin 119871infin (0 119879S119899)

119877119894 (sdot) 119877119894 (sdot) isin 119871infin (0 119879S119898119894) 119866119894 119866119894 isin S119899

(5)

and there exists a constant 120575 gt 0 such that for ae 119905 isin [0 119879]119876119894 (119905) 119876119894 (119905) + 119876119894 (119905) 119878119894 (119905) 119878119894 (119905) + 119878119894 (119905) ge 0119877119894 (119905) 119877119894 (119905) + 119877119894 (119905) ge 120575119868119866119894 119866119894 + 119866119894 ge 0

(6)

For the sake of notation simplicity the time argument issuppressed in cost functional above and in the sequel of thispaper wherever necessary

Let us now explain the mean-filed backward stochasticStackelberg differential game system In the game Player 1is the follower and Player 2 is the leader In addition forany 119894 = 1 2 we assume that 119869119894(120585 1199061 1199062) is a cost functionalfor Player 119894 Therefore at the initial time for any giventerminal target 120585 isin 1198712F119879(ΩR119899) the leader announces hisstrategy 1199062 isin U2[0 119879] over the whole planning horizon[0 119879] Then with the knowledge of the leaderrsquos strategy thefollower determines his response strategy 1199061(sdot) isin U1[0 119879]over the entire horizon to minimize 1198691(120585 1199061 1199062) Since thefollowerrsquos optimal response can be determined by the leaderthe leader can take it into account in finding and announcingher optimal strategy which minimizes 1198692(120585 1199061 1199062) over 1199062 isinU2[0 119879] In a little more rigorous way we give the followingdefinition of the Stackelberg game

Definition 1 The pair (1199061 1199062) isin U1[0 119879] timesU2[0 119879] is calledan open-loop solution to the above Stackelberg game if itsatisfies the following conditions

(i) There exists a map Γ U2[0 119879] times 1198712F119879(ΩR119899) 997888rarrU1[0 119879] such that1198691 (120585 Γ (1199062 120585) 1199062) = min

1199061isinU1[0119879]1198691 (120585 1199061 1199062)

forall1199062 isin U2 [0 119879] (7)

(ii) There exists a unique 1199062 isin U2[0 119879] such that1198692 (120585 Γ (1199062 120585) 1199062) = min1199062isinU2[0119879]

1198692 (120585 Γ (1199062 120585) 1199062) (8)

(iii) The optimal strategy of the follower is 1199061 = Γ(1199062 120585)The aim of the paper is to find the feedback represen-

tation of the open-loop Stackelberg strategy for mean-fieldbackward differential game (1) and (4)

3 Optimization for the Follower

In this section we consider the optimization problem of thefollower For given 1199062 isin U2[0 119879] the follower wants to solvethe following LQ optimal control problems for mean-fieldbackward stochastic differential system

4 Mathematical Problems in Engineering

Problem (BMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199061 isinU1[0 119879] such that

1198691 (120585 1199061 1199062) = min1199061isinU1[0119879]

1198691 (120585 1199061 1199062) (9)

By using the similar method found in [21] we are able toobtain the following result

Proposition 2 Under (H1)-(H2) let the terminal state 120585 isin1198712F119905(ΩR119899) and the leaderrsquos strategy 1199062 isin U2[0 119879] be given(i) The problem (BMF-LQ) is uniquely solvable with(119910 119911 1199061) being the only optimal pair if and only if there exists

a unique 4-tuple (119909 119910 119911 1199061) satisfying the MF-FBSDE

119889119909 = (119860⊤119909 + 119860⊤E119909 + 1198761119910 + 1198761E119910)119889119905 + (119862⊤119909+ 119862⊤E119909 + 1198781119911 + 1198781E119911) 119889119882(119905)

119889119910 = minus (119860119910 + 119860E119910 + 11986111199061 + 1198611E1199061 + 11986121199062 + 1198612E1199062+ 119862119911 + 119862E119911) 119889119905 + 119911119889119882 (119905) 119909 (0) = 1198661119910 (0) + 1198661E119910 (0) 119910 (119879) = 120585

(10)

and the following stationarity condition holds

11987711199061 + 1198771E1199061 + 119861⊤1 119909 + 119861⊤1 E119909 = 0 (11)

(2) The following relations are satisfied

E119910 = minus1198752E119909 minus E120601119910 minus E119910 = minus1198751 (119909 minus E119909) minus (120601 minus E120601) E119911 = minus [119868 + 1198751 (1198781 + 1198781)]minus1 [1198751 (119862 + 119862)⊤ E119909 + E120578] 119911 minus E119911 = minus (119868 + 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578]

(12)

where 1198751 1198752 and (120601 120578) are the solutions of the following threeequations respectively

1 + 1198601198751 + 1198751119860⊤ minus 119875111987611198751 + 1198611119877minus11 119861⊤1 + 119862 (119868+ 11987511198781)minus1 1198751119862⊤ = 01198751 (119879) = 0(13)

2 + (119860 + 119860)1198752 + 1198752 (119860 + 119860)⊤ + (1198611 + 1198611) (1198771+ 1198771)minus1 (1198611 + 1198611)⊤ + (119862 + 119862) [119868 + 1198751 (1198781 + 1198781)]minus1sdot 1198751 (119862 + 119862)⊤ minus 1198752 (1198761 + 1198761) 1198752 = 01198752 (119879) = 0(14)

119889120601 = minus [(119860 minus 11987511198761) (120601 minus E120601)+ (119860 + 119860 minus 1198752 (1198761 + 1198761))E120601 minus 11986121199062 minus 1198612E1199062+ 119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 E120578] 119889119905+ 120578119889119882 (119905) 120601 (119879) = minus120585

(15)

In above proposition we get the optimal strategy

1199061 = minus119877minus11 119861⊤1 (119909 minus E119909) minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ E119909 (16)

of the follower with any given leaderrsquos strategy 1199062 isin U2[0 119879]through the adjoint equation 119909 (one part of the Hamiltoniansystem (10)-(11)) Next we intend to obtain the state feedbackform of 1199061 We first have the following lemma

Lemma 3 Under (H1)-(H2) let (119909 119910 119911) be the solution of theHamiltonian system (10)-(11) Then

119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593 (17)

where 1198753 and 1198754 satisfy the following two Riccati equationsrespectively

3 minus 119860⊤1198753 minus 1198761 minus 1198753119860 + 11987531198611119877minus11 119861⊤1 1198753+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤1198753 = 01198753 (0) = 119866(18)

4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤) 1198754 minus (1198761 + 1198761) = 01198754 (0) = 1198661 + G1(19)

and 120593 is given by the following MF-SDE

119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578 119889119882(119905) 120593 (0) = 0

(20)

Mathematical Problems in Engineering 5

where 119860119910 ≜ 119860 + 1198611199101198753119860119910 ≜ 119860 + 119860 + 1198611199101198754119861119910 ≜ minus1198611119877minus11 119861⊤1 119861119910 ≜ minus (119861 + 1198611) (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ 119860120601 ≜ 119860 minus 11987511198761119860120601 ≜ 119860 + 119860 minus 1198752 (1198761 + 1198761) 119862120601 ≜ 119862 (119868 + 11987511198781)minus1 119862120601 ≜ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 119860120593 ≜ [119860119910 minus 1198621206011198751119862⊤1198753]⊤ 119860120593 ≜ [119860119910 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987531198751)minus1 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987541198752)minus1 119863120593 ≜ (1198781 minus 1198753) (119868 + 11987511198781)minus1 119863120593 ≜ (1198781 + 1198781 minus 1198753) (119868 + 1198751 (1198781 + 1198781))minus1

(21)

Proof Since (119909 119910 119911) is the solution of the Hamiltoniansystem (10)-(11) by noticing relations (12) we can get that theinitial value of 119909 satisfies

E119909 (0) = minus (1198661 + 1198661) 1198752 (0)E119909 (0)minus (1198661 + 1198661)E120601 (0) 119909 (0) minus E119909 (0) = minus11986611198751 (0) (119909 (0) minus E119909 (0))minus 1198661 (120601 (0) minus E120601 (0)) (22)

By using the fact that(119868 + 119872119873)minus1119872 = 119872(119868 + 119873119872)minus1 forall119873 119872 isin S119899 (23)

where we assume that 119868 + 119873119872 and 119868 + 119872119873 are reversible itcan be shown by a straightforward computation that119909 (0)= minus1198661 (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))

minus (1198661 + 1198661) [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (24)

which implies119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (25)

Therefore noting (11) and (21) we can rewrite the backwardstochastic differential equation 119910 of FBDSE (10) as thefollowing form MF-SDE

119889119910 = minus 119860119910 + 119860E119910 + (119861119910 minus 1198621206011198751119862⊤) (119909 minus E119909)+ (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909 minus 119862120601 (120578 minus E120578)minus 119862120601E120578 + 11986121199062 + 1198612E1199062 119889119905 minus (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ E119909 + E120578] 119889119882 (119905)

119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0)) minus [119868+ 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0)

(26)

Then we conjecture that 119909 and 119910 are related by the following119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593E119909 = 1198754E119910 + E120593119909 minus 119864119909 = 1198753 (119910 minus E119910) + 120593 minus E120593 (27)

where 1198753 1198754 [0 119879] 997888rarr S119899 are absolutely continuous withinitial value 1198661 1198661 + 1198661 respectively and 120593 satisfies119889120593 = 120572119889119905 + 120573119889119882 (119905) 120593 (0) = 0 (28)

for some F119905-progressively measurable processes 120572 and 120573Note that E119909 and E119910 satisfy the following two ordinarydifferential equations respectively

119889E119909 = [(119860 + 119860)⊤ E119909 + (1198761 + 1198761)E119910] 119889119905119909 (0) = (1198661 + 1198661)E119910 (0) 119889E119910 = minus (119860 + 119860)E119910 + (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909minus 119862120601E120578 + (1198612 + 1198612)E1199062 119889119905

E119910 (0) = minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (29)

Applying Itorsquos formula to the second equation of (27) we get

0 = [4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)1198754 minus (1198761 + 1198761)]E119910+ E120572 minus [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593+ 1198754119862120601E120578 minus 1198754 (1198612 + 1198612)E1199062

(30)

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 4: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

4 Mathematical Problems in Engineering

Problem (BMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199061 isinU1[0 119879] such that

1198691 (120585 1199061 1199062) = min1199061isinU1[0119879]

1198691 (120585 1199061 1199062) (9)

By using the similar method found in [21] we are able toobtain the following result

Proposition 2 Under (H1)-(H2) let the terminal state 120585 isin1198712F119905(ΩR119899) and the leaderrsquos strategy 1199062 isin U2[0 119879] be given(i) The problem (BMF-LQ) is uniquely solvable with(119910 119911 1199061) being the only optimal pair if and only if there exists

a unique 4-tuple (119909 119910 119911 1199061) satisfying the MF-FBSDE

119889119909 = (119860⊤119909 + 119860⊤E119909 + 1198761119910 + 1198761E119910)119889119905 + (119862⊤119909+ 119862⊤E119909 + 1198781119911 + 1198781E119911) 119889119882(119905)

119889119910 = minus (119860119910 + 119860E119910 + 11986111199061 + 1198611E1199061 + 11986121199062 + 1198612E1199062+ 119862119911 + 119862E119911) 119889119905 + 119911119889119882 (119905) 119909 (0) = 1198661119910 (0) + 1198661E119910 (0) 119910 (119879) = 120585

(10)

and the following stationarity condition holds

11987711199061 + 1198771E1199061 + 119861⊤1 119909 + 119861⊤1 E119909 = 0 (11)

(2) The following relations are satisfied

E119910 = minus1198752E119909 minus E120601119910 minus E119910 = minus1198751 (119909 minus E119909) minus (120601 minus E120601) E119911 = minus [119868 + 1198751 (1198781 + 1198781)]minus1 [1198751 (119862 + 119862)⊤ E119909 + E120578] 119911 minus E119911 = minus (119868 + 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578]

(12)

where 1198751 1198752 and (120601 120578) are the solutions of the following threeequations respectively

1 + 1198601198751 + 1198751119860⊤ minus 119875111987611198751 + 1198611119877minus11 119861⊤1 + 119862 (119868+ 11987511198781)minus1 1198751119862⊤ = 01198751 (119879) = 0(13)

2 + (119860 + 119860)1198752 + 1198752 (119860 + 119860)⊤ + (1198611 + 1198611) (1198771+ 1198771)minus1 (1198611 + 1198611)⊤ + (119862 + 119862) [119868 + 1198751 (1198781 + 1198781)]minus1sdot 1198751 (119862 + 119862)⊤ minus 1198752 (1198761 + 1198761) 1198752 = 01198752 (119879) = 0(14)

119889120601 = minus [(119860 minus 11987511198761) (120601 minus E120601)+ (119860 + 119860 minus 1198752 (1198761 + 1198761))E120601 minus 11986121199062 minus 1198612E1199062+ 119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 E120578] 119889119905+ 120578119889119882 (119905) 120601 (119879) = minus120585

(15)

In above proposition we get the optimal strategy

1199061 = minus119877minus11 119861⊤1 (119909 minus E119909) minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ E119909 (16)

of the follower with any given leaderrsquos strategy 1199062 isin U2[0 119879]through the adjoint equation 119909 (one part of the Hamiltoniansystem (10)-(11)) Next we intend to obtain the state feedbackform of 1199061 We first have the following lemma

Lemma 3 Under (H1)-(H2) let (119909 119910 119911) be the solution of theHamiltonian system (10)-(11) Then

119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593 (17)

where 1198753 and 1198754 satisfy the following two Riccati equationsrespectively

3 minus 119860⊤1198753 minus 1198761 minus 1198753119860 + 11987531198611119877minus11 119861⊤1 1198753+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤1198753 = 01198753 (0) = 119866(18)

4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤) 1198754 minus (1198761 + 1198761) = 01198754 (0) = 1198661 + G1(19)

and 120593 is given by the following MF-SDE

119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578 119889119882(119905) 120593 (0) = 0

(20)

Mathematical Problems in Engineering 5

where 119860119910 ≜ 119860 + 1198611199101198753119860119910 ≜ 119860 + 119860 + 1198611199101198754119861119910 ≜ minus1198611119877minus11 119861⊤1 119861119910 ≜ minus (119861 + 1198611) (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ 119860120601 ≜ 119860 minus 11987511198761119860120601 ≜ 119860 + 119860 minus 1198752 (1198761 + 1198761) 119862120601 ≜ 119862 (119868 + 11987511198781)minus1 119862120601 ≜ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 119860120593 ≜ [119860119910 minus 1198621206011198751119862⊤1198753]⊤ 119860120593 ≜ [119860119910 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987531198751)minus1 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987541198752)minus1 119863120593 ≜ (1198781 minus 1198753) (119868 + 11987511198781)minus1 119863120593 ≜ (1198781 + 1198781 minus 1198753) (119868 + 1198751 (1198781 + 1198781))minus1

(21)

Proof Since (119909 119910 119911) is the solution of the Hamiltoniansystem (10)-(11) by noticing relations (12) we can get that theinitial value of 119909 satisfies

E119909 (0) = minus (1198661 + 1198661) 1198752 (0)E119909 (0)minus (1198661 + 1198661)E120601 (0) 119909 (0) minus E119909 (0) = minus11986611198751 (0) (119909 (0) minus E119909 (0))minus 1198661 (120601 (0) minus E120601 (0)) (22)

By using the fact that(119868 + 119872119873)minus1119872 = 119872(119868 + 119873119872)minus1 forall119873 119872 isin S119899 (23)

where we assume that 119868 + 119873119872 and 119868 + 119872119873 are reversible itcan be shown by a straightforward computation that119909 (0)= minus1198661 (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))

minus (1198661 + 1198661) [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (24)

which implies119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (25)

Therefore noting (11) and (21) we can rewrite the backwardstochastic differential equation 119910 of FBDSE (10) as thefollowing form MF-SDE

119889119910 = minus 119860119910 + 119860E119910 + (119861119910 minus 1198621206011198751119862⊤) (119909 minus E119909)+ (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909 minus 119862120601 (120578 minus E120578)minus 119862120601E120578 + 11986121199062 + 1198612E1199062 119889119905 minus (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ E119909 + E120578] 119889119882 (119905)

119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0)) minus [119868+ 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0)

(26)

Then we conjecture that 119909 and 119910 are related by the following119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593E119909 = 1198754E119910 + E120593119909 minus 119864119909 = 1198753 (119910 minus E119910) + 120593 minus E120593 (27)

where 1198753 1198754 [0 119879] 997888rarr S119899 are absolutely continuous withinitial value 1198661 1198661 + 1198661 respectively and 120593 satisfies119889120593 = 120572119889119905 + 120573119889119882 (119905) 120593 (0) = 0 (28)

for some F119905-progressively measurable processes 120572 and 120573Note that E119909 and E119910 satisfy the following two ordinarydifferential equations respectively

119889E119909 = [(119860 + 119860)⊤ E119909 + (1198761 + 1198761)E119910] 119889119905119909 (0) = (1198661 + 1198661)E119910 (0) 119889E119910 = minus (119860 + 119860)E119910 + (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909minus 119862120601E120578 + (1198612 + 1198612)E1199062 119889119905

E119910 (0) = minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (29)

Applying Itorsquos formula to the second equation of (27) we get

0 = [4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)1198754 minus (1198761 + 1198761)]E119910+ E120572 minus [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593+ 1198754119862120601E120578 minus 1198754 (1198612 + 1198612)E1199062

(30)

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 5: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 5

where 119860119910 ≜ 119860 + 1198611199101198753119860119910 ≜ 119860 + 119860 + 1198611199101198754119861119910 ≜ minus1198611119877minus11 119861⊤1 119861119910 ≜ minus (119861 + 1198611) (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ 119860120601 ≜ 119860 minus 11987511198761119860120601 ≜ 119860 + 119860 minus 1198752 (1198761 + 1198761) 119862120601 ≜ 119862 (119868 + 11987511198781)minus1 119862120601 ≜ (119862 + 119862) (119868 + 1198751 (1198781 + 1198781))minus1 119860120593 ≜ [119860119910 minus 1198621206011198751119862⊤1198753]⊤ 119860120593 ≜ [119860119910 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987531198751)minus1 119862120593 ≜ (119868 + 11987531198751) 119862⊤120601 (119868 + 11987541198752)minus1 119863120593 ≜ (1198781 minus 1198753) (119868 + 11987511198781)minus1 119863120593 ≜ (1198781 + 1198781 minus 1198753) (119868 + 1198751 (1198781 + 1198781))minus1

(21)

Proof Since (119909 119910 119911) is the solution of the Hamiltoniansystem (10)-(11) by noticing relations (12) we can get that theinitial value of 119909 satisfies

E119909 (0) = minus (1198661 + 1198661) 1198752 (0)E119909 (0)minus (1198661 + 1198661)E120601 (0) 119909 (0) minus E119909 (0) = minus11986611198751 (0) (119909 (0) minus E119909 (0))minus 1198661 (120601 (0) minus E120601 (0)) (22)

By using the fact that(119868 + 119872119873)minus1119872 = 119872(119868 + 119873119872)minus1 forall119873 119872 isin S119899 (23)

where we assume that 119868 + 119873119872 and 119868 + 119872119873 are reversible itcan be shown by a straightforward computation that119909 (0)= minus1198661 (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))

minus (1198661 + 1198661) [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (24)

which implies119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0))minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (25)

Therefore noting (11) and (21) we can rewrite the backwardstochastic differential equation 119910 of FBDSE (10) as thefollowing form MF-SDE

119889119910 = minus 119860119910 + 119860E119910 + (119861119910 minus 1198621206011198751119862⊤) (119909 minus E119909)+ (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909 minus 119862120601 (120578 minus E120578)minus 119862120601E120578 + 11986121199062 + 1198612E1199062 119889119905 minus (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ E119909 + E120578] 119889119882 (119905)

119910 (0) = minus (119868 + 1198751 (0) 1198661) (120601 (0) minus E120601 (0)) minus [119868+ 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0)

(26)

Then we conjecture that 119909 and 119910 are related by the following119909 = 1198753 (119910 minus E119910) + 1198754E119910 + 120593E119909 = 1198754E119910 + E120593119909 minus 119864119909 = 1198753 (119910 minus E119910) + 120593 minus E120593 (27)

where 1198753 1198754 [0 119879] 997888rarr S119899 are absolutely continuous withinitial value 1198661 1198661 + 1198661 respectively and 120593 satisfies119889120593 = 120572119889119905 + 120573119889119882 (119905) 120593 (0) = 0 (28)

for some F119905-progressively measurable processes 120572 and 120573Note that E119909 and E119910 satisfy the following two ordinarydifferential equations respectively

119889E119909 = [(119860 + 119860)⊤ E119909 + (1198761 + 1198761)E119910] 119889119905119909 (0) = (1198661 + 1198661)E119910 (0) 119889E119910 = minus (119860 + 119860)E119910 + (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)E119909minus 119862120601E120578 + (1198612 + 1198612)E1199062 119889119905

E119910 (0) = minus [119868 + 1198752 (0) (1198661 + 1198661)]minus1 E120601 (0) (29)

Applying Itorsquos formula to the second equation of (27) we get

0 = [4 minus (119860 + 119860)⊤ 1198754 minus 1198754 (119860 + 119860)minus 1198754 (119861119910 minus 1198621206011198751 (119862 + 119862)⊤)1198754 minus (1198761 + 1198761)]E119910+ E120572 minus [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593+ 1198754119862120601E120578 minus 1198754 (1198612 + 1198612)E1199062

(30)

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Mathematical Problems in Engineering

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Page 6: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

6 Mathematical Problems in Engineering

and it implies that Riccati equation 1198754 is given by (19) and E120572satisfies

E120572 = [119860 + 119860 + 1198611199101198754 minus 1198621206011198751 (119862 + 119862)⊤ 1198754]⊤ E120593minus 1198754119862120601E120578 + 1198754 (1198612 + 1198612)E1199062 (31)

and similarly by applying the Itorsquos formula to the thirdequation of (27) we can obtain the following equation

0 = 119862⊤ (119909 minus E119909) + (119862 + 119862)⊤ E119909 + (1198753 minus 1198781) (119868+ 11987511198781)minus1 [1198751119862⊤ (119909 minus E119909) + 120578 minus E120578] + [1198753minus (1198781 + 1198781)] (119868 + 1198751 (1198781 + 1198781))minus1 [1198751 (119862 + 119862)⊤ E119909+ E120578] minus 120573 119889119882(119905) + [3 minus 1198753119860 minus 119860⊤1198753minus 1198761 + 1198753 (1198611119877minus11 119861⊤1 + 119862 (119868 + 11987511198781)minus1 1198751119862⊤) 1198753]sdot (119910 minus E119910) + (11987531198611119877minus11 119861⊤1+ 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤) (120593 minus E120593)minus 11987531198612 (1199062 minus E1199062) + 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ (120572 minus E120572) 119889119905

(32)

Hence 1198753 should be a solution of Riccati equation (18) and120572 120573 should satisfy120573 = (119868 + 11987531198751) (119868 + 11987811198751)minus1 119862⊤ (119909 minus E119909) + (119868 + 11987531198751)sdot (119868 + (1198781 + 1198781) 1198751)minus1 (119862 + 119862)⊤ E119909 + (1198753 minus 1198781)sdot (119868 + 11987511198781)minus1 (120578 minus E120578) + [1198753 minus (1198781 + 1198781)]sdot (119868 + 1198751 (1198781 + 1198781))minus1 E120578(33)

120572 minus E120572= minus (11987531198611119877minus11 119861⊤1 + 1198753119862 (119868 + 11987511198781)minus1 1198751119862⊤ minus 119860⊤)sdot (120593 minus E120593) minus 1198753119862 (119868 + 11987511198781)minus1 (120578 minus E120578)+ 11987531198612 (1199062 minus E1199062) (34)

Finally substituting (12) into (27) it is easy to show thefollowing119909 minus 119864119909 = minus (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]

E119909 = minus (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) (35)

Thus noticing (31) (33) and (34) we get that 120601 is given byMF-SDE (20)

Remark 4 In Lemma 3 to get the relation between 119909 and119910 we introduce another two Riccati equations which have

a unique solution respectively (see [20 Section 4]) Since(20) is a linear MF-SDEwith bounded coefficients and squareintegrable nonhomogeneous terms it has a unique solution 120593(see [13 Section 2])

Based on the above lemma we obtain the main conclu-sion of problem (BMF-LQ)

Theorem 5 Under (H1)-(H2) Problem (BMF-LQ) is solvablewith the optimal open-loop strategy 1199061 being of a feedbackrepresentation1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593]

minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (1198754E119910 + E120593) (36)

where 1198753 1198754 and 120593 are the solutions of (18) (19) and (20)respectively The optimal state trajectory (119910 119911) is the uniquesolution of the MF-BSDE119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)

+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882(119905) 119910 (119879) = 120585(37)

Moreover the optimal cost of the follower is

1198811 (120585 1199062 1199061) = Eint1198790[10038171003817100381710038171003817(119868 + 11987511198753)minus1 [1198751 (120593 minus E120593)

+ (120601 minus E120601)]1003817100381710038171003817100381721198761 + 10038171003817100381710038171003817(119868 + 11987521198754)minus1 (1198752E120593+ E120601)1003817100381710038171003817100381721198761+1198761 + 10038171003817100381710038171003817119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)]100381710038171003817100381710038172119877minus1

1

+ 100381710038171003817100381710038171003817(1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)1003817100381710038171003817100381710038172(1198771+1198771)minus1 + 10038171003817100381710038171003817(119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]1003817100381710038171003817100381721198781 + 100381710038171003817100381710038171003817[119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593)minus E120578]10038171003817100381710038171003817100381721198781+1198781] 119889119905 + 1003817100381710038171003817(119868 + 1198751 (0) 1198661) (120601 (0)minus E120601 (0))100381710038171003817100381721198661 + 100381710038171003817100381710038171003817[119868 + 1198752 (0) (1198661 + 1198661)]minus1sdot E120601 (0)1003817100381710038171003817100381710038172119866+1198661

(38)

Here (120601 120578) is the solution of MF-BSDE (15)

Proof The first assertion is the direct consequence of Propo-sition 2 and Lemma3 For the second assertion since 1199061 given

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 7: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 7

by (36) is the optimal strategy of the follower with terminalcondition 120585 and the leaderrsquos strategy 1199062 which are given wehave that the optimal cost of the follower is as follows1198811 (120585 1199062 1199061) ≜ 1198691 (120585 1199061 1199062)

= Eint1198790[⟨1198761 (119910 minus E119910) 119910 minus E119910⟩

+ ⟨(1198761 + 1198761)E119910E119910⟩+ ⟨1198771 (1199061 minus E1199061) 1199061 minus E1199061⟩+ ⟨(1198771 + 1198771)E1199061E1199061⟩ + ⟨1198781 (119911 minus E119911) 119911 minus E119911⟩+ ⟨(1198781 + 1198781)E119911E119911⟩] 119889119905 + ⟨1198661 (119910 (0)minus E119910 (0)) 119910 (0) minus E119910 (0)⟩ + ⟨(119866 + 1198661)E119910 (0) E119910 (0)⟩

(39)

Noting (12) (17) and (36) we get

E119910 = minus (119868 + 11987521198754)minus1 (1198752E120593 + E120601) 119910 minus E119910 = minus (119868 + 11987511198753)minus1 [1198751 (120593 minus E120593) + (120601 minus E120601)] E119911 = [119868 + 1198751 (1198781 + 1198781)]minus1sdot [1198751 (119862 + 119862)⊤ (119868 + 11987541198752)minus1 (1198754E120601 minus E120593) minus E120578] 119911 minus E119911 = (119868 + 11987511198781)minus1sdot [1198751119862⊤ (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601) minus (120593 minus E120593)]minus (120578 minus E120578)]

1199061 minus E1199061 = 119877minus11 119861⊤1 (119868 + 11987531198751)minus1 [1198753 (120601 minus E120601)minus (120593 minus E120593)] E1199061 = (1198771 + 1198771)minus1 (1198611 + 1198611)⊤ (119868 + 11987541198752)minus1 (1198754E120601minus E120593)

(40)

and substitution of the above into (85) completes the proof

Remark 6 If 1198612 = 1198612 = 0 ie there is only one player inthis game this game problem degrades into the same as thatstudied by [21] However in [21] they did not get the feedbackform open-loop optimal control which is given byTheorem 5in our paper

Remark 7 In Theorem 5 we get the feedback form optimalcontrol of the follower by (36) and it is easy to see thatthe optimal control 1199061 is a functional about the leaderrsquoscontrol 1199062 Furthermore if the leader announces his control1199062 the follower should choose his map Γ U2[0 119879] times

1198712F119879(ΩR119899) 997888rarr U1[0 119879] as the form of (36) to minimizethe cost functional 11986914 Optimization for the Leader

In above section we get the open-loop feedback form optimalcontrol of problem (BMF-LQ) for any given 120585 and 1199062 Nowlet problem (BMF-LQ) be uniquely solvable for any given(120585 1199062) isin 1198712F119879(ΩR119899) timesU2[0 119879] Since the followerrsquos optimalresponse 1199061 of form (36) can be determined by the leader theleader can take it into account in finding and announcing hisoptimal strategy Consequently the leader has the followingstate equation119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578) minus 1198754119862120601E120578 119889119905+ 119862120593 (120593 minus E120593) + 119862120593E120593 minus 1198621205931198753 (120601 minus E120601)

minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578) minus 119863120593E120578 119889119882(119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882 (119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593) + 119861119910E120593+ 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062 + 119862 (119911 minus E119911)+ (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 120585120601 (119879) = minus120585

(41)

with the coefficients given by (21) It should be mentionedthat the ldquostaterdquo in (41) is the quintuple (120593 120601 120578 119910 119911) Since(41) is a decoupled ldquo1 times 2rdquo MF-FBSDE (one forward and twobackward mean-field type stochastic differential equations)the solvability for F119905119905ge0-adapted solution (120593 120601 120578 119910 119911) canbe easily guaranteed The leader would like to choose hiscontrol such that his cost functional

1198692 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[⟨1198762119910 119910⟩

+ ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩ + ⟨1198772E1199062E1199062⟩+ ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905 + ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(42)

is minimizedThe optimal control problem for the leader canbe stated as follows

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 8: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

8 Mathematical Problems in Engineering

Problem (FBMF-LQ) For given 120585 isin 1198712F119879(ΩR119899) find a 1199062 isinU2[0 119879] such that1198692 (120585 1199062) = min

1199062isinU2[0119879]1198692 (120585 1199062) (43)

Remark 8 In the traditional leader-follower game for (mean-field) forward stochastic differential equations the stateprocesses of leaders are all given by an ldquo1 times 1rdquo (MF) FBSDEHowever in backward case the state process of the leaderbecomes a ldquo1 times 2rdquo MF-FBSDE Thus the problem we studiedis more complex and technically challenging

Under (H1)-(H2) by noting [23 proposition 1] and [21Theorem 22] we get that the cost functional is strictly convexand coercive which means that problem (FBMF-LQ) has aunique optimal control Then we will use the variationalmethod to solve the (FBMF-LQ)

Proposition 9 Let (H1)-(H2) hold Let (120593 120601 120578 119910 119911 1199062) be theoptimal sextuplet for the terminal state 120585 isin 1198712F119879(ΩR119899) Thenthe solution (1199091 1199092 1199101 1199111) to the MF-FBSDE1198891199091 = [119860⊤119910 (1199091 minus E1199091) + 119860⊤119910E1199091 + 1198762119910 + 1198762E119910] 119889119905+ [119862⊤ (1199091 minus E1199091) + (119862 + 119862)⊤ E1199091 + 1198782119911

+ 1198782E119911] 119889119882(119905) 1198891199092 = [119860⊤120601 (1199092 minus E1199092) + 119860⊤120601E1199092 minus 1198753119862⊤120593 (1199111 minus E1199111)minus 1198754119862⊤120593E1199111] 119889119905 + [119862⊤120601 (1199092 minus E1199092) + 119862⊤120601E1199092minus 119862⊤1206011198753 (1199101 minus E1199101) minus 119862⊤1206011198754E1199101 minus 119863⊤120593 (1199111 minus E1199111)minus 119863⊤120593E1199111] 119889119882(119905) 1198891199101 = minus [119860⊤120593 (1199101 minus E1199101) + 119860⊤120593E1199101 + 119862⊤120593 (1199111 minus E1199111)+ 119862⊤120593E1199111 + 119861⊤119910 (1199091 minus E1199091) + 119861⊤119910E1199091] 119889119905+ 1199111119889119882(119905) 1199091 (0) = 1198662119910 (0) + 1198662E119910 (0) 1199092 (0) = 01199101 (119879) = 0

(44)

satisfies119861⊤2 1199091 + 119861⊤2 E1199091 minus 119861⊤2 1199092 minus 119861⊤2 E1199092 + 119861⊤2 1198753 (1199101 minus E1199101)+ (1198612 + 1198612)⊤ 1198754E1199101 + 11987721199062 + 1198772E1199062 = 0 (45)

Proof For any 1199062 isin U2[0 119879] and any 120598 isin R let (120593 120601 120578 119910 119911)be the solution of119889120593 = 119860120593 (120593 minus E120593) + 119860120593E120593 + 11987531198612 (1199062 minus E1199062)+ 1198754 (1198612 + 1198612)E1199062 minus 1198753119862120601 (120578 minus E120578)

minus 1198754119862120601E120578 119889119905 + 119862120593 (120593 minus E120593) + 119862120593E120593minus 1198621205931198753 (120601 minus E120601) minus 1198621205931198754E120601 minus 119863120593 (120578 minus E120578)minus 119863120593E120578119889119882 (119905) 119889120601 = minus [119860120601 (120601 minus E120601) + 119860120601E120601 minus 1198612 (1199062 minus E1199062)minus (1198612 + 1198612)E1199062 + 119862120601 (120578 minus E120578) + 119862120601E120578] 119889119905+ 120578119889119882(119905) 119889119910 = minus [119860119910 (119910 minus E119910) + 119860119910E119910 + 119861119910 (120593 minus E120593)+ 119861119910E120593 + 1198612 (1199062 minus E1199062) + (1198612 + 1198612)E1199062+ 119862 (119911 minus E119911) + (119862 + 119862)E119911] 119889119905 + 119911119889119882 (119905) 120593 (0) = 0119910 (119879) = 0120601 (119879) = 0

(46)

Let (120593120598 120601120598 120578120598 119910120598 119911120598) be the solution to the perturbedstate equation then it is clear that (120593120598 120601120598 120578120598 119910120598 119911120598) =(120593 120601 120578 119910 119911) + 120598(120593 120601 120578 119910 119911) and hence

0 = lim120598997888rarr0

1198692 (120585 1199062 + 1205981199062) minus 1198692 (120585 1199062)120598= Eint119879

0[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨11987721199062 1199062⟩

+ ⟨1198772E1199062E1199062⟩ + ⟨1198782119911 119911⟩ + ⟨1198782E119911E119911⟩] 119889119905+ ⟨1198662119910 (0) 119910 (0)⟩ + ⟨1198662E119910 (0) E119910 (0)⟩

(47)

Noting that

Eint1198790[⟨119873 (120583 minus E120583) ]⟩ minus ⟨119873⊤ (] minus E]) 120583⟩] 119889119905

= Eint1198790[⟨119873120583 ]⟩ minus ⟨119873⊤] 120583⟩ minus ⟨119873E120583 ]⟩

+ ⟨119873⊤E] 120583⟩] 119889119905 = 0forall119873 isin R119899times119898 ] isin 1198712F119905 (0 119879R119899) 120583 isin 1198712F119905 (0 119879R119898)

(48)

and applying Itorsquos formula to ⟨1199091 119910⟩ + ⟨1199092 120601⟩ minus ⟨1199101 120593⟩ wehave

Eint1198790[⟨1199062 minus E1199062 119861⊤2 (11987531199101 + 1199091 minus 1199092)⟩

+ ⟨E1199062 (1198612 + 1198612)⊤ (11987541199101 + 1199091 minus 1199092)⟩] 119889119905= Eint1198790[⟨1198762119910 119910⟩ + ⟨1198762E119910E119910⟩ + ⟨1198782119911 119911⟩

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 9: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 9

+ ⟨1198782E119911E119911⟩] 119889119905 + E ⟨1198662119910 (0) 119910 (0)⟩+ ⟨1198662E119910 (0) E119910 (0)⟩ (49)

Noting (47) and the fact that Eint1198790⟨E1199062 1199101⟩119889119905 = Eint119879

0⟨1199062

E1199101⟩119889119905 we have0 = Eint119879

0[⟨1199062

119861⊤2 [1198753 (1199101 minus E1199101) + (1199091 minus E1199091) minus (1199092 minus E1199092)]+ (1198612 + 1198612)⊤ (1198754E1199101 + E1199091 minus E1199092) + 11987721199062+ 1198772E1199062⟩] 119889119905(50)

which implies (45)

Here denote that

119883 ≜ (11990911199092120593) 119884 ≜ ( 1199101206011199101)119885 ≜ ( 1199111205781199111)

B1 ≜ ( 1198612minus11986120 ) B1 ≜ ( 1198612 + 1198612minus1198612 minus 11986120 ) A ≜ diag (119860119910 119860120601 119860⊤120593) A ≜ diag (119860119910 119860120601 119860⊤120593) C ≜ diag (119862 119862120601 119862⊤120593) C ≜ diag (119862 + 119862 119862120601 119862⊤120593) Q ≜ diag (1198762 0 0) Q ≜ diag (1198762 + 1198762 0 0)

B2 ≜ ( 0011987531198612)B2 ≜ ( 001198754 (1198612 + 1198612)) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) D ≜ ( 0 0 1198611199100 0 0119861⊤119910 0 0 ) F ≜ (0 0 00 0 minus1198753119862⊤1205930 minus1198753119862120601 0 ) F ≜ (0 0 00 0 minus1198754119862⊤1205930 minus1198754119862120601 0 ) S ≜ (1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) S ≜ (1198782 + 1198782 0 00 0 minus119863⊤1205930 minus119863120593 0 ) G ≜ diag (1198662 0 0) G ≜ diag (1198662 + 1198662 0 0) 120585 = ( 120585minus1205850 )

(51)

From the above result we see that if 1199062 happens to be theoptimal control of problem (MF-FBSLQ) for terminal state120585 isin 1198712F119879(ΩR119899) then the following MF-FBSDE admits anadapted solution (119883119884 119885)119889119884 = minus [A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +C (119885 minus E119885) +CE119885 +D (119883 minus E119883)+DE119883]119889119905 + 119885119889119882 (119905)

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Page 10: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

10 Mathematical Problems in Engineering

119889119883 = [A⊤ (119883 minus E119883) +A⊤E119883 +B2 (1199062 minus E1199062)+B2E1199062 + Q (119884 minus E119884) + QE119884 +F (119885 minus E119885)

+FE119885]119889119905 + [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 +S (119885 minus E119885)

+ SE119885] 119889119882(119905) 119883 (0) = G (119884 (0) minus E119884 (0)) + GEY (0) 119884 (119879) = 120585

(52)

And the following stationarity condition holds

B⊤1 (119883 minus E119883) +B

⊤

1 E119883 +B⊤2 (119884 minus E119884) +B

⊤

2 E119884+ 11987721199062 + 1198772E1199062 = 0 (53)

We now use the idea of the four-step scheme (see [21 27 28])to study the solvability of the above MF-FBSDE (52)

Remark 10 Another thing which we should keep in mindis that different with the traditional forward LQ leader-follower game whose Hamiltonian system is a 2 times 2-FBSDE(or MF-FBSDE) with terminal conditions coupling in ourbackward system case the Hamiltonian system that we getis a 3 times 3-MF-FBSDE with initial conditions coupling sothat to decouple the corresponding Hamiltonian system andget the feedback form optimal control we should introducefour Riccati type equations with more higher dimension andcomplex coupling

Suppose we have the relation119884 = minusΠ1 (119883 minus E119883) minus Π2E119883 minus 120601 (54)

Namely

E119884 = minusΠ2E119883 minus E120601119884 minus E119884 = minusΠ1 (119883 minus E119883) minus (120601 minus E120601) (55)

Here Π1 Π2 [0 119879] 997888rarr S3119899 are absolutely continuous withterminal condition Π1(0) = Π2(0) = 0 and 120601 satisfies thefollowing MF-BSDE119889120601 = minus119889119905 + 120578119889119882 (119905) 120601 (119879) = minus120585 (56)

where is some F119905-progressively measurable processes tobe confirmed Here we should point out that even thoughthere are no nonhomogeneous terms in the MF-FBSDE(52) since the influence of the initial terms coupling weshould also introduce the nonhomogeneous term 120601 in thepossible connection (54) between 119884 and 119883 and it is the

essential differencewith traditional caseNote that (taking themathematical expectation in (52)-(53))

119889E119884 = minus [AE119884 +B1E1199062 +CE119885 +DE119883]119889119905119889E119883 = [A⊤E119883 +B2E1199062 + QE119884 +FE119885] 119889119905E119883(0) = GE119884 (0) E119884 (119879) = E120585B⊤

1 E119883 +B⊤

2 E119884 + (1198772 + 1198772)E1199062 = 0(57)

Thus

119889 (119884 minus E119884) = minus [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 + 119885119889119882 (119905) 119889 (119883 minus E119883) = [A⊤ (119883 minus E119883) +B2 (1199062 minus E1199062)+ Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ [C⊤ (119883 minus E119883) +C

⊤E119883 +F

⊤ (119884 minus E119884)+F⊤E119884 + S (119885 minus E119885) + SE119885] 119889119882(119905) 119883 (0) minus E119883(0) = G (119884 (0) minus E119884 (0)) 119884 (119879) minus E119884 (119879) = 120585 minus E120585

B⊤1 (119883 minus E119883) +B

⊤2 (119884 minus E119884) + 1198772 (1199062 minus E1199062) = 0

(58)

Applying Itorsquos formula to the first equation of (55) and noting(55) and (58) we have

0 = minus119889 (119884 minus E119884) minus Π1 (119883 minus E119883)119889119905 minus Π1119889 (119883 minus E119883)minus 119889 (120601 minus E120601) = [A (119884 minus E119884) +B1 (1199062 minus E1199062)+C (119885 minus E119885) +D (119883 minus E119883)] 119889119905 minus 119885119889119882 (119905)minus Π1 (119883 minus E119883)119889119905 minus Π1 [A⊤ (119883 minus E119883)+B2 (1199062 minus E1199062) + Q (119884 minus E119884) +F (119885 minus E119885)] 119889119905+ ( minus E) 119889119905 minus 120578119889119882(119905) minus Π1 [C⊤ (119883 minus E119883)+C⊤E119883 +F

⊤ (119884 minus E119884) +F⊤E119884 +S (119885 minus E119885)

+SE119885] 119889119882(119905) = minus (A minusB1119877minus12 B⊤2+ Π1B2119877minus12 B⊤2 minus Π1Q) [Π1 (119883 minus E119883) + 120601 minus E120601]+ (C minus Π1F) (119885 minus E119885) minus (Π1 + Π1A⊤minus Π1B2119877minus12 B⊤1 minusD +B1119877minus12 B⊤1 ) (119883 minus E119883) + (

Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering 11

minus E) 119889119905 minus Π1 [C⊤ (119883 minus E119883) +C⊤E119883

+F⊤ (119884 minus E119884) +F

⊤E119884 + S (119885 minus E119885) + SE119885]

+ 119885 + 120578119889119882 (119905) (59)

This implies (assuming that 119868 + Π1S and 119868 + Π1S areinvertible)

E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π2F)⊤ E119883minus Π1F⊤E120601 + E120578]

119885 minus E119885 = minus (119868 + Π1S)minus1 [Π1 (C minus Π1F)⊤ (119883 minus E119883)minus Π1F⊤ (120601 minus E120601) + 120578 minus E120578]

(60)

Substitution of (60) into (59) now gives minus E = (C minus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + [AminusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601) (61)

Π1 + (A minusB1119877minus12 B⊤2 )Π1 + Π1 (A minusB1119877minus12 B⊤2 )⊤+ Π1 (B2119877minus12 B⊤2 minus Q)Π1 + (C minus Π1F) (119868+ Π1S)minus1Π1 (C minus Π1F)⊤ +B1119877minus12 B⊤1 minusD = 0Π1 (119879) = 0(62)

Similarly applying Itorsquos formula to the second equation of(55) and noting (55) (57) we have0 = minus (A minusB1 (1198772 + 1198772)minus1B⊤2

+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Q)E120601 + (Cminus Π2F)E119885 + [minusAΠ2 +B1 (1198772 + 1198772)minus1B⊤2Π2minus Π2 (B2 (1198772 + 1198772)minus1B⊤2 minus Q)Π2 minus Π2 +D

minusB1 (1198772 + 1198772)minus1B⊤1 minus Π2A⊤+ Π2B2 (1198772 + 1198772)minus1B⊤1 ]E119883 + E

(63)

This implies (noting (60))

E120572 = [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π2F) (119868 + Π1S)minus1 E120578

(64)

Π2 + [A minusB1 (1198772 + 1198772)minus1B⊤2 ]Π2+ Π2 [B2 (1198772 + 1198772)minus1B⊤2 minus Q]Π2 +B1 (1198772+ 1198772)minus1B⊤1 minusD + Π2 [AminusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ + (C minus Π2F) (119868+ Π1S)minus1Π1 (C minus Π2F)⊤ = 0Π2 (119879) = 0

(65)

Noticing (61) and (64) we obtain

119889120601 = minus [A minusB1119877minus12 B⊤2 + Π1B2119877minus12 B⊤2 minus Π1Qminus (C minus Π1F) (119868 + Π1S)minus1Π1F⊤] (120601 minus E120601)+ [A minusB1 (1198772 + 1198772)minus1B⊤2+ Π2B2 (1198772 + 1198772)minus1B⊤2 minus Π2Qminus (C minus Π2F) (119868 + Π1S)minus1Π1F⊤]E120601 + (Cminus Π1F) (119868 + Π1S)minus1 (120578 minus E120578) + (C minus Π2F) (119868+ Π1S)minus1 E120578 119889119905 + 120578119889119882 (119905)

120601 (119879) = minus120585

(66)

Then to get the feedback representation of the leader weshould try to give another connection between 119883 and 119884Namely suppose we have the following relation119883 = Π3 (119884 minus E119884) + Π4E119884 + 120593 (67)

ie

E119883 = Π4E119884 + E120593119883 minus E119883 = Π3 (119884 minus E119884) + (120593 minus E120593) (68)

Here Π3 Π4 [0 119879] 997888rarr S3119899 are absolutely continuous and 120593satisfies the following MF-SDE119889120593 = 120573119889119905 + 120574119889119882 (119905) 120593 (0) = 0 (69)

where 120573 and 120574 are some F119905-progressively measurable pro-cesses to be confirmed In addition noting (55) and (68) wehave

E119883 = minus (119868 + Π4Π2)minus1 (Π4E120601 minus E120593) 119883 minus E119883= minus (119868 + Π3Π1)minus1 [Π3 (120601 minus E120601) minus (120593 minus E120593)] (70)

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Mathematical Problems in Engineering

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Page 12: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

12 Mathematical Problems in Engineering

Thus by noticing (60) the process 119885 is given by

E119885 = C120593E120593 +C120601E120601 +C120578E120578119885 minus E119885 = C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) (71)

where

C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π2F)⊤ (119868 + Π4Π2)minus1 C120601 = minusC120578Π1 (Π4C +F)⊤ (119868 + Π2Π4)minus1F⊤C120578 = minus (119868 + Π1S)minus1 C120593 = C120578Π1 (C minus Π1F)⊤ (119868 + Π3Π1)minus1 C120601 = minusC120578Π1 (Π3C +F)⊤ (119868 + Π1Π3)minus1

(72)

Then by using the similar method in proving Lemma 3 wecan get that the initial value of 119884 is given by119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0))

minus (119868 + Π2G)minus1 E120601 (0) (73)

Thus noting (60) the process 119884 satisfies the following MF-SDE119889119884 = minus A (119884 minus E119884) +AE119884 +B1 (1199062 minus E1199062)+B1E1199062 +D (119883 minus E119883) +DE119883+C [C120593 (120593 minus E120593) + C120601 (120601 minus E120601) + C120578 (120578 minus E120578)]

+C (C120593E120593 +C120601E120601 +C120578E120578) 119889119905 + C120593E120593+C120601E120601 +C120578E120578 + C120593 (120593 minus E120593) + C120601 (120601 minus E120601)+ C120578 (120578 minus E120578) 119889119882 (119905)

119884 (0) = minus (119868 + Π1G)minus1 (120601 (0) minus E120601 (0)) minus (119868+ Π2G)minus1 E120601 (0)

(74)

and applying Itorsquos formula to the first equation of (68) andnoting (57) and (71) we have0= (A⊤ + Π4D)E119883minus (Π4B1 +B2) (1198772 + 1198772)minus1 (B⊤1 E119883 +B

⊤

2 E119884)+ (Q minus Π4 + Π4A)E119884+ (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578) minus E120573(75)

It implies that (noting (68))

Π4 = [A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤Π4+ (Q minusB2 (1198772 + 1198772)minus1B⊤2 )+ Π4 [A minusB1 (1198772 + 1198772)minus1B⊤2 ]+ Π4 [D minusB1 (1198772 + 1198772)minus1B⊤1 ]Π4Π4 (0) = G

(76)

E120573= [A⊤ + Π4D minus (Π4B1 +B2) (1198772 + 1198772)minus1B⊤1 ]sdot E120593 + (Π4C +F) (C120593E120593 +C120601E120601 +C120578E120578)

(77)

Similarly applying Itorsquos formula to the second equation of(68) and noting (58) and (71) we obtain

Π3 = (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 ) + Π3 (D minusB1119877minus12 B⊤1 )Π3Π3 (0) = G(78)

120573 minus E120573 = [A⊤ minusB2119877minus12 B⊤1 + Π3 (D minusB1119877minus12 B⊤1 )+ (F + Π3C) C120593] (120593 minus E120593) + (F + Π3C)sdot [C120578 (120578 minus E120578) + C120601 (120601 minus E120601)] (79)

120574 = (119868 + Π3Π1) (119868 +SΠ1)minus1 (C minus Π1F)⊤ (119868+ Π3Π1)minus1 (120593 minus E120593) + (119868 + Π3Π1) (119868 +SΠ1)minus1 (Cminus Π2F)⊤ (119868 + Π4Π2)minus1 E120593 minus (119868 + Π3Π1) (119868+SΠ1)minus1 (Π3C +F)⊤ (119868 + Π1Π3)minus1 (120601 minus E120601)minus (119868 + Π3Π1) (119868 +SΠ1)minus1 (Π4C +F)⊤ (119868+ Π2Π4)minus1 E120601 + (Π3 minusS) (119868 + Π1S)minus1 (120578 minus E120578)+ (Π3 minusS) (119868 + Π1S)minus1 E120578

(80)

Therefore 120593 satisfies the following MF-SDE119889120593 = (120573 minus E120573 + E120573) 119889119905 + 120574119889119882(119905) 120593 (0) = 0 (81)

where 120573 minus E120573 E120573 and 120574 are given by (79) (77) and (80)respectively Then we get the main result of the section

Theorem 11 Under (H1)-(H2) suppose Riccati equations (62)(65) (78) (76) admit differentiable solution Π1 Π2 Π3 Π4

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Mathematical Problems in Engineering

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Page 13: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 13

respectively Problem (FBMF-LQ) is solvable with the optimalopen-loop strategy 1199062 being of a feedback representation1199062 = minus119877minus12 (Π3B1 +B2)⊤ (119884 minus E119884)

minus (1198772 + 1198772)minus1 (Π4B1 +B2)⊤ E119884minus 119877minus12 B⊤1 (120593 minus E120593) minus (1198772 + 1198772)minus1B⊤1 E120593(82)

where Π3 Π4 and 120593 are the solutions of (78) (76) and (81)respectively The optimal state trajectory (119884 119885) is the uniquesolution of the MF-BSDE119889119884 = minus [(A +DΠ3) (119884 minus E119884) + (A +DΠ4)E119884+B1 (1199062 minus E1199062) +B1E1199062 +C (119885 minus E119885) +CE119885+D (120593 minus E120593) +DE120593] 119889119905 + 119885119889119882 (119905) 119884 (119879) = 120585

(83)

Moreover the optimal cost of the follower is

1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062) = Eint1198790[10038171003817100381710038171003817(119868 + Π1Π3)minus1

sdot [(120601 minus E120601) + Π1 (120593 minus E120593)]100381710038171003817100381710038172Q + 10038171003817100381710038171003817(119868 + Π2Π4)minus1sdot (E120601 + Π2E120593)100381710038171003817100381710038172Q + 10038171003817100381710038171003817(Π3B1 +B2)⊤ (119868+ Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] minusB⊤1 (120593minus E120593)1003817100381710038171003817100381721198772 + 100381710038171003817100381710038171003817(Π4B1 +B2)⊤ (119868 + Π2Π4)minus1 (E120601

+ Π2E120593) minusB⊤

1 E1205931003817100381710038171003817100381710038172(1198772+1198772) + 10038171003817100381710038171003817C120593 (120593 minus E120593)+ C120601 (120601 minus E120601) + C120578 (120578 minus E120578)100381710038171003817100381710038172S2 + 10038171003817100381710038171003817C120593E120593+C120601E120601 +C120578E120578100381710038171003817100381710038172S2] 119889119905 + 10038171003817100381710038171003817(119868 + Π1G)minus1sdot (120601 (0) minus E120601 (0))100381710038171003817100381710038172G + 100381710038171003817100381710038171003817(119868 + Π2G)minus1 E120601 (0)1003817100381710038171003817100381710038172G

(84)

where S2 = diag(1198781 0 0) and S2 = diag(1198781 + 1198781 0 0)Proof Firstly by noting (45) and (67) we get the feedbackrepresentation (82) of the leaderrsquos optimal strategy 1199062 withldquostaterdquo 119884 For the second assertion we have that the optimalcost of the leader is1198812 (120585 1199062) ≜ 1198692 (120585 1199061 1199062)

= Eint1198790[⟨Q (119884 minus E119884) 119884 minus E119884⟩ + ⟨QE119884E119884⟩

+ ⟨1198772 (1199062 minus E1199062) 1199062 minus E1199062⟩+ ⟨(1198772 + 1198772)E1199062E1199062⟩

+ ⟨S2 (119885 minus E119885) 119885 minus E119885⟩ + ⟨S2E119885E119885⟩] 119889119905+ ⟨G (119884 (0) minus E119884 (0)) 119884 (0) minus E119884 (0)⟩+ ⟨GE119884 (0) E119884 (0)⟩

(85)

Noting (70) and (55) we get the following

E119884 = minus (119868 + Π2Π4)minus1 (E120601 + Π2E120593) 119884 minus E119884 = minus (119868 + Π1Π3)minus1 [(120601 minus E120601) + Π1 (120593 minus E120593)] (86)

Substitution (71) (73) (82) and (86) into (85) completes theproof

Here we wanted to emphasize that for generality weonly assume that the REs used in above theorem haveunique solutions respectively Furthermore we will presentsome sufficient conditions for the solvability of them in theappendix

Remark 12 If we let 119860 = 1198611 = 1198612 = 119862 = 119876119894 = 119877119894 = 119878119894 =119866119894 = 0 119894 = 1 2 then 1198751 = 1198752 1198753 = 1198754 Π1 = Π2 Π3 = Π4and the game problem studied in this paper will degrade intothe problem of the LQ leader-follower game for BSDE As weknow it is not studied before

Likewise noting (36) and (68) the optimal control 1199061 ofthe follower can also be represented in a similar way as 1199062 in(82)

1199061 = minus119877minus11 119861⊤1 [1198753 (119910 minus E119910) + 120593 minus E120593] minus (1198771 + 1198771)minus1sdot (1198611 + 1198611)⊤ (1198754E119910 + E120593)= minus119877minus11 119861⊤1 [(1198753 0 0) (119884 minus E119884)+ (0 0 119868) (119883 minus E119883)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 0)E119884 + (0 0 119868)E119883]= minus119877minus11 119861⊤1 [(1198753 0 Π3) (119884 minus E119884)+ (0 0 119868) (120593 minus E120593)] minus (1198771 + 1198771)minus1 (1198611 + 1198611)⊤sdot [(1198754 0 Π4)E119884 + (0 0 119868)E120593]

(87)

On the existence and uniqueness of the open-loop Stackel-berg strategy we have the following conclusion

Theorem 13 Under (H1)-(H2) If (62) (65) (78) (76) admit atetrad of solutions Π1 Π2 Π3 Π4 then the open-loop Stackel-berg strategy exists and is unique In this case the unique open-loop Stackelberg strategy in feedback representation is (1199061 1199062)given by (87) and (82) In addition the optimal costs of thefollower and the leader are given by (38) and (84) respectively

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

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Mathematical Problems in Engineering

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Page 14: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

14 Mathematical Problems in Engineering

5 Application to Pension Fund Problemand Simulation

In this section we present an LQ Stackelberg game for MF-BSDE of the defined benefit (DB) pension fund It is wellknown that defined benefit (DB) pension scheme is one oftwo main categories In a DB scheme there are two corre-sponding representative members who make contributionscontinuously over time to the pension fund in [0 119879] One ofthe members is the leader (ie the supervisory governmentor company) with the regular premium proportion 1199062 andthe other one is the follower (ie individual producer orretail investor) with the regular premium proportion 1199061Premiums decided by two members are payable in regularpremiums ie premiums are a proportion of salary whichare continuously deposited into the planmemberrsquos individualaccount See [8 22 29ndash31] for more details

Now consider the one-dimension investment problem(ie 119899 = 1) We can invest two tradable assets a risk-freeasset given by the following ODE1198891198780 (119905) = 119877 (119905) 1198780 (119905) 119889119905 (88)

where119877(119905) is the interest rate at time 119905 the other one is a stockwith price satisfying linear SDE119889119878 (119905) = 119878 (119905) [120583 (119905) 119889119905 + 120590 (119905) 119889119882 (119905)] (89)

where 120583(119905) is its instantaneous rate of return and 120590(119905) isits instantaneous volatility Then the value process 119910(119905) ofpension fund plan memberrsquos account is governed by thedynamic119889119910 (119905)= 119910 (119905) [119877 (119905) + (120583 (119905) minus 119877 (119905)) 120587 (119905)] 119889119905 + 120590120587119889119882 (119905)+ [1199061 (119905) + 1199062 (119905)] 119889119905 (90)

where 120587 is the portfolio process On the one hand if thepension fund manager wants to achieve the wealth level 120585 atthe terminal time 119879 to fulfill hisher obligations and on theother hand if we set 119911 = 120590120587119910 then the above equation isequivalent to

119889119910 (119905) = [119877 (119905) 119910 (119905) + (120583 (119905) minus 119877 (119905))120590 (119905) 119911 (119905) + 1199061 (119905)+ 1199062 (119905)] 119889119905 + 119911119889119882 (119905)

119910 (119879) = 120585(91)

where 1199061(sdot) 1199062(sdot) are two control variables correspondingto the regular premium proportion of two agents For anyagents it is natural to hope that the process of wealthrsquosvariance var(119910(119905)) is as small as possible Therefore theminimized cost functionals are revised as119869119894 (120585 1199061 1199062) = Eint119879

0[var (119910 (119905)) + 1199062119894 (119905)] 119889119905

= Eint1198790[1199102 (119905) minus (E119910 (119905))2 + 1199062119894 (119905)] 119889119905 (92)

0

02

04

06

08

1

12

14

16

18

2

Value

P1P2

P3P4

01 02 03 04 05 06 07 08 09 10Time

Figure 1 The solution curve of Riccati equations 1198751 1198752 1198753 1198754

Π1(1 1)

Π1(1 2)

Π1(1 3)

Π1(2 2)

Π1(2 3)

Π1(3 3)

01 02 03 04 05 06 07 08 09 10Time

minus06

minus04

minus02

0

02

04

06

08

1

12Va

lue

Figure 2 The solution curve of Riccati equationsΠ1In addition if we let 119877(119905) = 01 120583(119905) = 05 120590(119905) = 04 119879 =1 120585 = 10 + 119882(119879) we can get the value of the correspondingRiccati equations used in our paper In addition by usingthe Eulerrsquos method we plot the solution curves of all Riccatiequations which are shown in Figures 1 2 3 and 4

Here we should point out that Riccati equations 119875119895 119895 =1 2 3 4 are one-dimension functions and given by Figure 1Furthermore for any 119895 = 1 2 3 4 sinceΠ119895 is 3times3 symmetricmatrix function we have that Π119895(119898 119899) = Π119895(119899119898) whereΠ119895(119898 119899) means the value of the 119898 row and the 119899 columnof Π119895 In addition in Figure 4 Π3(1 2) = Π3(2 2) =Π3(2 3) = 0 Therefore there are only four lines in Figure 4Next by noting (76) (66) and (81) we have that Π4 (120601 120578)and 120593 are given by the following equations or Figure 5respectively

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 15

0 1090807060504030201Time

Π2(1 1)Π2(1 2)Π2(1 3)

Π2(2 2)Π2(2 3)Π2(3 3)

minus2

minus15

minus1

minus05

0

05

1

15

2

Value

Figure 3The solution curve of Riccati equations Π2

Π3(1 1)

Π3(1 2)

Π3(1 3)

Π3(2 2)

Π3(2 3)

Π3(3 3)

02 04 06 08 10Time

minus02

minus01

0

01

02

03

04

05

06

07

Value

Figure 4 The solution curve of Riccati equationsΠ3Π4 = 0120578 = E120578 = (minus1 1 0)⊤ E120593 = 0 (93)

Applying Theorem 13 we can obtain the open-loopStackelberg strategy with feedback representation of twoagents in the pension fund problem studied in this section

(1)

(2)

(3)

minus15

minus10

minus5

0

5

10

15

Value

02 04 06 08 10Time

Figure 5 The solution curve of 1206016 Conclusion

We study the open-loop Stackelberg strategy for LQ mean-field backward stochastic differential game Both the cor-responding two mean-field stochastic LQ optimal controlproblems for follower and follower have been discussed Byvirtue of eight REs two MF-SDEs and two MF-BSDEs theStackelberg equilibrium has been represented as the feedbackform involving the state as well as its mean Based on thatas an application of the LQ Stackelberg game of mean-field backward stochastic differential system in financialmathematics a class of stochastic pension fund optimizationproblems with two representative members is discussed Ourpresent work suggests various future research directionsfor example (i) to study the backward Stackelberg gamewith indefinite control weight (this will formulate the mean-variance analysis with relative performance in our setting)and (ii) to study the backwardmean-field game in Stackelbergstrategy (this will involve 119873 followers rather than one inthe game) We plan to study these issues in our futureworks

Appendix

In this section we concentrate on the solvability of the REsΠ119894(sdot) 119894 = 1 2 3 4 which are used in Theorem 11 Forsimplicity we will consider only the constant coefficient caseNow we firstly consider the solvability of Π3(sdot) (the solutionof (78)) Noting that it is given the initial value of the RE (78)by making the time reversing transformation

120591 = 119879 minus 119905 119905 isin [0 119879] (A1)

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

16 Mathematical Problems in Engineering

we obtain that the RE (78) is equivalent to

Π3 + (A minusB1119877minus12 B⊤2 )⊤Π3 + Π3 (A minusB1119877minus12 B⊤2 )+ (Q minusB2119877minus12 B⊤2 )+ Π3 (D minusB1119877minus12 B⊤1 )Π3 = 0Π3 (119879) = G(A2)

Then we introduce the following Riccati equation

Ξ3 + Ξ3ΦΞ31 + [ΦΞ31]⊤ Ξ3 + Ξ3 (D minusB1119877minus12 B⊤1 ) Ξ3+ ΦΞ32 = 0Ξ3 (119879) = 0(A3)

where ΦΞ31 ≜ (A minus B1119877minus12 B⊤2 ) + (D minus B1119877minus12 B⊤1 )G andΦΞ32 ≜ (A minus B1119877minus12 B⊤2 )⊤G + G(A minus B1119877minus12 B⊤2 ) + (Q minusB2119877minus12 B⊤2 ) + G(D minus B1119877minus12 B⊤1 )G Furthermore it is easyto see that solution Π3(sdot) of (78) and that Ξ3(sdot) of (A3) arerelated by the following

Π3 (119905) = G + Ξ3 (119905) 119905 isin [0 119879] (A4)

Next we let

AΞ3 = ( ΦΞ31 D minusB1119877minus12 B⊤1minusΦΞ32 minus [ΦΞ31]⊤ ) (A5)

Then according to [27 Chapter 2] we have the followingconclusion and representation of Π3(sdot)Proposition 14 Let (H1)-(H2) hold and let det(0119868)119890AΞ3 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (A3) admits a unique so-lution Ξ3(sdot) which has the following representationΞ3 (119905)= minus[(0 119868) 119890AΞ3 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ3 (119879minus119905) (1198680) 119905 isin [0 119879]

(A6)

Moreover (A4) gives the solution Π3(sdot) of the Riccati equation(78)

In addition using the samemethodwe can get the similarconclusion about Π4(sdot) of (76)Proposition 15 Let (H1)-(H2) hold and let det(0119868)119890AΞ4 119905 ( 0119868 ) gt 0 119905 isin [0 119879] Then (78) admits a unique solu-tion Π4(sdot) which has the following representation

Π4 (119905)= G

minus [(0 119868) 119890AΞ4 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ4 (119879minus119905) (1198680) 119905 isin [0 119879] (A7)

Here AΞ4 = ( ΦΞ41 DminusB1(1198772+1198772)minus1B⊤

1

minusΦΞ42

minus[ΦΞ41]⊤

) ΦΞ41 ≜ A minus B1(1198772 +1198772)minus1B⊤2 +[DminusB1(1198772+1198772)minus1B⊤1 ]G andΦΞ42 ≜ [AminusB1(1198772+1198772)minus1B⊤2 ]⊤G + G[A minus B1(1198772 + 1198772)minus1B⊤2 ] + Q minus B2(1198772 +1198772)minus1B⊤2 +G[D minusB1(1198772 + 1198772)minus1B⊤1 ]G

In the rest of this section we concentrate on the solvabilityof Π1(sdot) and Π2(sdot) with the case 119862 = 119862 = 0 By noting thedefinitions (21) it is easy to get that C = F = C = F = 0Then similar to what we did in the previous proof we get thefollowing proposition

Proposition 16 Let (H1)-(H2) hold and 119862 = 119862 = 0 Inaddition let det (0 119868)119890AΞ119894 119905 ( 0119868 ) gt 0 119905 isin [0 119879] 119894 = 1 2Then (62) and (65) admit unique solutions Π1(sdot) and Π2(sdot)which have the following representationΠ119894 (119905)

= minus [(0 119868) 119890AΞ119894 (119879minus119905) (0119868)]minus1 (0 119868) 119890AΞ119894 (119879minus119905) (1198680) 119905 isin [0 119879] 119894 = 1 2(A8)

respectively where

AΞ1 ≜ ((A minusB1119877minus12 B⊤2 )⊤ B2119877minus12 B⊤2 minus Q

D minusB1119877minus12 B⊤1 B1119877minus12 B⊤2 minusA)

AΞ2

≜ ([A minusB1 (1198772 + 1198772)minus1B⊤2 ]⊤ B2 (1198772 + 1198772)minus1B⊤2 minus Q

D minusB1 (1198772 + 1198772)minus1B⊤1 B1 (1198772 + 1198772)minus1B⊤2 minusA

) (A9)

Data Availability

All the numerical calculated data used to support the findingsof this study can be obtained by calculating the equations inthe paper and the codes used in this paper are available fromthe corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundationof China (no 11601285 no 61573217 and no 11831010) the

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 17: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Mathematical Problems in Engineering 17

National High-Level Personnel of Special Support Programand the Chang Jiang Scholar Program of Chinese EducationMinistry

References

[1] H V StackelbergMarktformUndGleichgewicht Springer 1934[2] T Basar and G J Olsder Dynamic Noncooperative Game

Theory vol 23 of Classics in Applied Mathematics Society forIndustrial and Applied Mathematics 1999

[3] N V Long A Survey of Dynamic Games in Economics WorldScientific Singapore 2010

[4] A Bagchi and T Basar ldquoStackelberg strategies in linear-quadratic stochastic differential gamesrdquo Journal of OptimizationTheory and Applications vol 35 no 3 pp 443ndash464 1981

[5] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002

[6] A Bensoussan S Chen and S P Sethi ldquoThe maximum prin-ciple for global solutions of stochastic Stackelberg differentialgamesrdquo SIAM Journal on Control and Optimization vol 53 no4 pp 1956ndash1981 2015

[7] B Oslashksendal L Sandal and J Uboslashe ldquoStochastic Stackelbergequilibria with applications to time-dependent newsvendormodelsrdquo Journal of Economic Dynamics amp Control vol 37 no7 pp 1284ndash1299 2013

[8] J Shi G Wang and J Xiong ldquoLeader-follower stochastic dif-ferential game with asymmetric information and applicationsrdquoAutomatica vol 63 pp 60ndash73 2016

[9] M Kac ldquoFoundations of kinetic theoryrdquo in Proceedings of theThird Berkeley Symposium on Mathematical Statistics and Prob-ability vol 3 Berkeley and Los Angeles California Universityof California Press 1956

[10] A Bensoussan J Frehse and P Yam Mean Field Games andMean Field Type Control Theory Springer Briefs in Mathemat-ics New York NY USA 2013

[11] M Huang P E Caines and R P Malhame ldquoLarge-populationcost-coupled LQG problems with non-uniform agentsindividual-mass behavior and decentralized 120576-nash equilibriardquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 52 no 9 pp 1560ndash1571 2007

[12] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[13] J Yong ldquoLinear-quadratic optimal control problems for mean-field stochastic differential equationsrdquo SIAM Journal on Controland Optimization vol 51 no 4 pp 2809ndash2838 2013

[14] Y Lin X Jiang andW Zhang ldquoOpen-loop stackelberg strategyfor the linear quadratic mean-field stochastic differential gamerdquoIEEE Transactions on Automatic Control 2018

[15] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[16] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[17] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoAnnals of Probability vol 37 no 4 pp 1524ndash1565 2009

[18] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquoMathematical Finance vol 7no 1 pp 1ndash71 1997

[19] M Kohlmann and X Y Zhou ldquoRelationship between back-ward stochastic differential equations and stochastic controlsa linear-quadratic approachrdquo SIAM Journal on Control andOptimization vol 38 no 5 pp 139ndash147 2000

[20] A E LimandXY Zhou ldquoLinear-quadratic control of backwardstochastic differential equationsrdquo SIAM Journal on Control andOptimization vol 40 no 2 pp 450ndash474 2001

[21] X Li J Sun and J Xiong ldquoLinear quadratic optimal controlproblems for mean-field backward stochastic differential equa-tionsrdquo Applied Mathematics amp Optimization pp 1ndash28 2017

[22] J Huang G Wang and J Xiong ldquoA maximum principle forpartial information backward stochastic control problems withapplicationsrdquo SIAM Journal on Control and Optimization vol48 no 4 pp 2106ndash2117 2009

[23] K Du J Huang and ZWu ldquoLinear quadratic mean-field-gameof backward stochastic differential systemsrdquoMathematical Con-trol amp Related Fields vol 8 pp 653ndash678 2018

[24] J Huang S Wang and Z Wu ldquoBackward mean-field linear-quadratic-gaussian (LQG) games full and partial informationrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 61 no 12 pp 3784ndash3796 2016

[25] J Shi and G Wang ldquoA nonzero sum differential game ofBSDE with time-delayed generator and applicationsrdquo Instituteof Electrical and Electronics Engineers Transactions onAutomaticControl vol 61 no 7 pp 1959ndash1964 2016

[26] G Wang and Z Yu ldquoA partial information non-zero sumdifferential game of backward stochastic differential equationswith applicationsrdquo Automatica vol 48 no 2 pp 342ndash352 2012

[27] J Ma and J Yong Forward-Backward Stochastic DifferentialEquations and Their Applications Lecture Notes in Mathemat-ics Springer Berlin Germany 1999

[28] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 of Applications of MathematicsSpringer New York NY USA 1999

[29] A J Cairns D Blake and K Dowd ldquoStochastic lifestylingoptimal dynamic asset allocation for defined contributionpension plansrdquo Journal of Economic Dynamics amp Control vol30 no 5 pp 843ndash877 2006

[30] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010

[31] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 18: Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential …downloads.hindawi.com/journals/mpe/2019/1798585.pdf · 2019-07-30 · information and gave some

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Dynamic Stackelberg Problems

Dynamic Stackelberg Problems

Pricing strategies (StackelBerg & Bertrand)

Pricing strategies (StackelBerg & Bertrand)

Sunk costs and contestability - Stackelberg - Spence - Dixit

Sunk costs and contestability - Stackelberg - Spence - Dixit

Stationary Strong Stackelberg Equilibrium in Discounted ...

Stationary Strong Stackelberg Equilibrium in Discounted ...

Cournot and Stackelberg equilibrium in mixed duopoly modelseprints.uanl.mx/7765/1/Cournot and Stackelberg... · 2015. 10. 8. · Cournot and Stackelberg equilibrium in mixed duopoly

Cournot and Stackelberg equilibrium in mixed duopoly modelseprints.uanl.mx/7765/1/Cournot and Stackelberg... · 2015. 10. 8. · Cournot and Stackelberg equilibrium in mixed duopoly

Stackelberg Routing in Arbitrary Networks

Stackelberg Routing in Arbitrary Networks

How Backward are the Other Backward Classes? Changing ... · How Backward are the Other Backward Classes? Changing Contours ... How Backward are the Other Backward ... hardly any

How Backward are the Other Backward Classes? Changing ... · How Backward are the Other Backward Classes? Changing Contours ... How Backward are the Other Backward ... hardly any

STACKELBERG BEATS COURNOT: ON COLLUSION AND …homepage.univie.ac.at/wieland.mueller/publications/...We report on an experiment designed to compare Stackelberg and Cournot duopoly

STACKELBERG BEATS COURNOT: ON COLLUSION AND …homepage.univie.ac.at/wieland.mueller/publications/...We report on an experiment designed to compare Stackelberg and Cournot duopoly

A Cournot-Stackelberg Model of Supply Contracts with Financial Hedgingmh2078/Master_Cournot-Stackelberg_R1.pdf · A Cournot-Stackelberg Model of Supply Contracts with Financial Hedging

A Cournot-Stackelberg Model of Supply Contracts with Financial Hedgingmh2078/Master_Cournot-Stackelberg_R1.pdf · A Cournot-Stackelberg Model of Supply Contracts with Financial Hedging

Stackelberg-Based Anti-Jamming Game for Cooperative ...

Stackelberg-Based Anti-Jamming Game for Cooperative ...

Cournot competition Bertrand competition Stackelberg ...schiu/oligo_slide.pdf · Cournot competition Bertrand competition Stackelberg competition Cournot Competition An industry is

Cournot competition Bertrand competition Stackelberg ...schiu/oligo_slide.pdf · Cournot competition Bertrand competition Stackelberg competition Cournot Competition An industry is

Trends and Applications in Stackelberg Security Gamesthanhhng/publications/... · Trends and Applications in Stackelberg Security Games Debarun Kar 1, Thanh H. Nguyen , Fei Fang ,

Trends and Applications in Stackelberg Security Gamesthanhhng/publications/... · Trends and Applications in Stackelberg Security Games Debarun Kar 1, Thanh H. Nguyen , Fei Fang ,

Stackelberg versus Cournot Oligopoly with Private …...Stackelberg versus Cournot Oligopoly with Private Information Eray Cumbuly IESE Business School, Barcelona Abstract In this

Stackelberg versus Cournot Oligopoly with Private …...Stackelberg versus Cournot Oligopoly with Private Information Eray Cumbuly IESE Business School, Barcelona Abstract In this

Multiobjective Mixed-Integer Stackelberg Gamescoral.ie.lehigh.edu/~ted/files/talks/LBMZSSPIC-EURO06.pdf · mixed-integer zero-sum static Stackelberg problem (MZSSP): MZSSP min x ...

Multiobjective Mixed-Integer Stackelberg Gamescoral.ie.lehigh.edu/~ted/files/talks/LBMZSSPIC-EURO06.pdf · mixed-integer zero-sum static Stackelberg problem (MZSSP): MZSSP min x ...

Comparing Human Behavior Models in Repeated Stackelberg ... · Several competing human behavior models have been proposed to model boundedly rational adversaries in repeated Stackelberg

Comparing Human Behavior Models in Repeated Stackelberg ... · Several competing human behavior models have been proposed to model boundedly rational adversaries in repeated Stackelberg

Achieving Network Optima Using Stackelberg Routing Strategies

Achieving Network Optima Using Stackelberg Routing Strategies

Stackelberg Model and Bayesian Games

Stackelberg Model and Bayesian Games

Stackelberg Plans · In our duopoly example 𝑡= 1𝑡, the time decision of the Stackelberg follower. Let 𝑡be a vector of decisions chosen by the Stackelberg leader at . The

Stackelberg Plans · In our duopoly example 𝑡= 1𝑡, the time decision of the Stackelberg follower. Let 𝑡be a vector of decisions chosen by the Stackelberg leader at . The

Multisensor Precipitation Estimator (MPE)In this presentation • What is MPE? • How has MPE come about? • What can MPE do now? • What are the upcoming improvements? MPE •

Multisensor Precipitation Estimator (MPE)In this presentation • What is MPE? • How has MPE come about? • What can MPE do now? • What are the upcoming improvements? MPE •

Sunk costs and contestability - Stackelberg - Spence - Dixitarchive.mu.ac.in/myweb_test/M.A.Part - I - Industrial Economics... · Sunk costs and contestability - Stackelberg - Spence

Sunk costs and contestability - Stackelberg - Spence - Dixitarchive.mu.ac.in/myweb_test/M.A.Part - I - Industrial Economics... · Sunk costs and contestability - Stackelberg - Spence