Research Article Nonzero-Sum Stochastic Differential Portfolio Games...

Research ArticleNonzero-Sum Stochastic Differential Portfolio Games undera Markovian Regime Switching Model

Chaoqun Ma1 Hui Wu1 and Xiang Lin2

1Business School of Hunan University Changsha 410082 China2School of Finance Zhejiang Gongshang University Hangzhou 310018 China

Correspondence should be addressed to Hui Wu h wu1018163com

Received 27 September 2014 Accepted 15 December 2014

Academic Editor Chuangxia Huang

Copyright copy 2015 Chaoqun Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a nonzero-sum stochastic differential portfolio game problem in a continuous-time Markov regime switchingenvironment when the price dynamics of the risky assets are governed by a Markov-modulated geometric Brownian motion(GBM) The market parameters including the bank interest rate and the appreciation and volatility rates of the risky assets switchover time according to a continuous-time Markov chain We formulate the nonzero-sum stochastic differential portfolio gameproblem as two utility maximization problems of the sum process between two investorsrsquo terminal wealth We derive a pair ofregime switching Hamilton-Jacobi-Bellman (HJB) equations and two systems of coupled HJB equations at different regimes Weobtain explicit optimal portfolio strategies and Feynman-Kac representations of the two value functions Furthermore we solvethe system of coupled HJB equations explicitly in a special case where there are only two states in the Markov chain Finally weprovide comparative statics and numerical simulation analysis of optimal portfolio strategies and investigate the impact of regimeswitching on optimal portfolio strategies

1 Introduction

The optimal portfolio selection has been studied extensivelyin modern finance This research is of great importance fromboth theoretical and practical purposesThe pioneering workcan be traced to Markowitz [1] which provided quantitativemethods to investigate a single-period optimal portfolioselection problem Then Merton [2 3] first extended thissingle-period model to a continuous-time model In theseminal work of Merton he obtained closed-form solutionsto the optimal portfolio selection problems Mertonrsquos workhas opened up an important field called continuous-timefinance We note that one of the key assumptions in Mertonrsquosoptimal portfolio models is that the model parameters areassumed to be constants and the price processes of riskyassets are modeled by the classical geometric Brownianmotions However we know that this key assumption is notconsistent with the actual behavior of asset price dynamicsEmpirical finance literature has found lots of stylized facts

in asset returns such as heavy tails in the asset returnsrsquodistributions time-varying volatility long-termmemory andregime switching Thus it would be of practical relevanceand importance to consider more realistic portfolio selectionmodels

In the past three decades or so among those establishedMertonrsquos portfolio models Markov regime switching anddifferential game models are two main extensions Regimeswitching models are an efficient and convenient approachto capture the cyclical features of structure changes inreal macroeconomic fundamentals Early works on regimeswitching models can be traced to Quandt [4] Hamilton [5]pioneered the econometric applications of Markov regimeswitching models Since then there has been a growinginterest in applications of regime switching models intofinance and economics Guo et al [6] built and solved a realoption model of investment decisions in which the growthrate and volatility of decision variable such as growth rateand diffusion coefficient shift between different states at

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 738181 18 pageshttpdxdoiorg1011552015738181

2 Mathematical Problems in Engineering

random times References [7ndash9] established a dynamic capitalstructure model and demonstrated how business-cycle varia-tions (regime switching) in expected growth rates economicuncertainty and risk premia influence firmsrsquo financing anddefault policies Some papers on optimal portfolio selectionunder regime switching models can refer to [10ndash16] andothers However it seems that one important issue that maybe overlooked by the existing literature on portfolio selectionis interactive decision making problems In this paper wedevelop amodel of two investors to study interactive decisionmaking on portfolio selection

The origin of the differential game theory might trace tothe 1940s In 1965 Isaacswrote the classical work ldquoDifferentialGamerdquo It has laid down the solid mathematical and theoret-ical foundations to the differential game theory Since thenstochastic differential game models have found a wide rangeof applications in finance Some early works include [17ndash19] and some references therein Some recent works include[20ndash24] Browne [20] formulated various versions of a zero-sum stochastic differential game to investigate dynamicoptimal investment problems between two ldquosmallrdquo investorsin continuous time He provided the existence conditions ofNash equilibrium and gave explicit representations for thevalue functions and optimal portfolio strategies Mannucci[25] studied Nash equilibrium for two-player nonzero-sumstochastic differential game Elliott and Siu [26] extended themodel to a continuous-timeMarkovian regime switching set-ting and continued to study the risk minimization portfolioselection problem by using stochastic differential game Siu[27] considered option pricing under regime switching by agame theoretical approach More recently Leong and Huang[28] developed a stochastic differential game of capitalism tostudy the role of uncertainty Lin [29] studied a nonzero-sumstochastic differential portfolio game between two investorsElliott and Siu [30] introduced a model which coveredeconomic risk financial risk insurance risk and model riskto discuss an optimal investment problem of an insurancecompany using stochastic differential game approach Liuand Yiu [31] considered stochastic differential games withVaR risk constraints between two insurance companiesTheyprovided explicit Nash equilibria and derived closed-formsolutions to value functions However It seems that the lit-erature has not well studied the optimal portfolio interactivedecision making problem under stochastic differential gamein a continuous-time Markovian regime switching setting

To the best of our knowledge our model is most relatedwith Browne [20] and Lin [29] since the two papers havediscussed stochastic differential portfolio games Howeverthe distinction between them and our paper is also evidentThere are two fundamental differences between the modelconsidered by them and the models considered here Firstlywe incorporate our model with a general continuous-timeMarkov regime switching setting and take into accountthe Markov regime switching risk and its impacts on thefinancial asset prices Secondly we consider a nonzero-sumstochastic differential portfolio game which is different fromthe several versions of zero-sum stochastic differential gamein Browne [20] Furthermore we use a stochastic optimalcontrol approach for the current nonzero-sum stochastic

differential portfolio game problem This method is differentfrom the approach used in Lin [29] namely stochastic linearquadratic control

In this paper we treat the optimal portfolio selectionproblem in a wide class of continuous-time Markovianregime switchingmodelsWe consider the portfolio selectionbetween two ldquosmallrdquo investors call them A and B (Theinvestors are called ldquosmallrdquo as their decision making behaviordoes not affect the market prices of the underlying assets)We consider a continuous-time financial market with threeprimitive securities namely a bank account and two riskyassets The dynamic price processes of all the primitivesecurities are assumed to be modulated by a continuous-time Markovian chain The rationale of using this regimeswitching model is to incorporate the impact of regime shiftson asset prices attributed to structure changes in marketor macroeconomy The two risky assets are correlated witheach other only one of which is available to each investorMoreover both investors are allowed to trade freely in thebank accountThe investors cooperate with each other by thechoice of their own portfolio strategies when they make deci-sions on investment We formulate the stochastic differentialgame as two utility maximization problems Two objectivefunctions are considered here One investor is trying tomaximize his payoff simultaneously the other investor actsantagonistically to maximize the other payoff Each payoffis formulated as expected utility of the wealth sum processof the two investors By using stochastic optimal controltheory we derive a pair of regime switchingHJB equations forthe value functions Moreover we obtain the Feynman-Kacrepresentations of value functions Closed-form expressionsfor optimal portfolio strategies are also obtained Finally wefind that Markov regime switching in the model parametershas a significant effect on the optimal portfolio strategies andvalue functions

Aside from the intrinsic probabilistic and game theoreticinterest such a model is applicable in many economicsettings Aswe know diversification improves the ability of aninvestorrsquos risk-return trade-off However it can be difficult fora small investor to hold enough stocks which are well diver-sified In addition maintaining a well-diversified portfoliocan lead to high transaction costs If several investors forma group well-diversified portfolio and low transaction costscan be realized Different investors have different attitudestowards risk so the choice of the stock and the goal ofinvestment for investors are different with each other

The rest of the paper is organized as followsThe followingsection presents the price and wealth dynamic processes ina continuous-time Markov regime switching economy InSection 3 we first introduce two optimal portfolio problemswith different objective functions And then we formulatea two-investor nonzero-sum stochastic differential portfoliogame problem In Section 4 we derive a pair of regimeswitching Hamilton-Jacobi-Bellman (HJB) equations for thenonzero-sum differential game problem and explicit solu-tions for the optimal portfolio strategies and value functionsof two investors are obtained In Section 5 we discuss onespecial case for a two-state Markov regime switching modelIn Section 6weprovide the comparative statics andnumerical

Mathematical Problems in Engineering 3

simulations analysis Finally we summarize the findings andoutline some potential topics for future research

2 Market Model

In this section we will consider a continuous-time Markovregime switching financial marketmodel consisting of a bankaccount and two risky assets (eg stocks or mutual funds)These assets are tradable continuously over a finite timehorizon [0 119879] where 119879 isin (0 infin) Denote the time horizon[0 119879] by T Same as [32] the standard assumptions offinancial market hold such as no transaction costs infinitelydivisible asset and information symmetric

A triple (ΩFP) is a probability space where Ω is aset F is a 120590-field of subsets of Ω and P is a real worldprobability measure on F A subset 119873 of Ω is negligibleif there exists 119861 isin F such that 119873 sub 119861 and P(119861) = 0The probability space is complete if F contains the set of allnegligible sets To model uncertainties that emerged in ourmodel we adopt a complete probability space with filtration(ΩF F

1199050le119905le119879

P) whereF = F119879and F

119905 describes the

flow of information available to investorsWe also assume theprobability space is rich enough to incorporate all sources ofrandomness arising from fluctuations of financial asset pricesand structural changes in macroeconomic conditions

Wemodel the evolution of the states of the economy overtime by a continuous-time finite state time-homogeneousobservable Markov chain 120585 = 120585(119905) | 119905 isin T defined on(ΩFP)with a finite state spaceS = s

1 s2 s

119873 sub R119873

where 119873 ge 2 The states of the Markov chain are interpretedas proxies of different observable macroeconomic indicatorssuch as gross domestic product (GDP) sovereign creditratings and consumer price index (CPI) More precisely wesuppose that the Markov chain is also right-continuous andirreducible

Without loss of generality following the convention of[33] we identify the state space of the chain as a finite set ofunit basis vectors E = e

1 e2 e

119873 where e

119894isin R119873 and

the 119895th component of e119894is the Kronecker delta denoted by 120575

119894119895

for each 119894 119895 = 1 2 119873 Kronecker delta 120575119894119895is a piecewise

function of variables 119894 and 119895 where 120575119894119895

= 1 if 119894 = 119895 otherwiseit is zero The set E is called the canonical representationof the state space of the Markov chain 120585 and it provides amathematically convenient way to represent the state spaceof the chain Here ldquo120585(119905) = e

119894rdquo means that macroeconomic

indicators are in state 119894 at time 119905To specify the statistics properties or the probability

law of the Markov chain we define stationary transitionprobabilities 119875

119894119895(119905) = P(120585(119905) = e

119895| 120585(0) = e

119894) for 119894 119895 =

1 2 119873 and 119905 ge 0 initial distribution 119901119894

= P(120585(0) = e119894)

and the generator 119876 = [119902119894119895

]119894119895=12119873

of the chain 120585 underPas follows

119902119894119895

=

lim119905rarr0+

119875119894119895

(119905)

119905119894 = 119895

lim119905rarr0+

119875119894119894

(119905) minus 1

119905119894 = 119895

119894 119895 = 1 2 119873 (1)

The generator 119876 is also called a rate matrix or a 119876-matrixHere for each 119894 119895 = 1 2 119873 119902

119894119895is the constant instan-

taneous intensity of the transition of the chain 120585 from statee119894to e119895 Note that 119902

119894119895ge 0 for 119894 = 119895 and sum

119873

119895=1119902119894119895

= 0so 119902119894119894

le 0 Here for each 119894 119895 = 1 2 119873 with 119894 = 119895 weassume that 119902

119894119895gt 0 So we obtain that 119902

119894119894lt 0 Then with the

canonical representation of the state space of the chain Elliottet al [33] provided the following semimartingale dynamicdecomposition for 120585

120585 (119905) = 120585 (0) + int119905

0

1198761015840120585 (119906) 119889119906 + M (119905) (2)

where 1015840 denotes the transpose of a matrix or a vector HereM(119905) | 119905 isin T is an R119873-valued martingale with respectto the filtration generated by Markov chain 120585 The filtrationF120585 = F120585(119905) | 119905 isin T satisfies the usual conditions whichare the right-continuousP-completed natural filtration

In what follows we will specify the price processes ofthe primitive securities and describe how the state of theeconomy represented by the chain 120585 influences the priceprocesses Note that the state space of the chain 120585 is a set ofunit basis vectors so any function of Markov chain 120585(119905) canbe denoted by a scalar product between a vector and 120585(119905)

Suppose 119903(119905) denote the instantaneous continuouslycompounded interest rate of the bank account at time 119905 foreach 119905 isin T Then the chain determines 119903(119905) as

119903 (119905) = 119903 (119905 120585 (119905)) = ⟨r 120585 (119905)⟩ (3)

where ⟨sdot sdot⟩ is the inner product in R119873 and r =

(1199031 1199032 119903

119873)1015840

isin R119873 with 119903119894

gt 0 for each 119894 = 1 2 119873119903119894is interpreted as the interest rate of the bank account when

the economy is in the 119894th state Here the inner product is todecide which component of the vectors of interest rate r driftrate 120583 or volatility rate 120590 is in force according to the state ofthe economy described by the chain 120585(119905) at a particular timeThen the price process of the bank account 119861 = 119861(119905) | 119905 isin

T evolves over time according to

119889119861 (119905) = 119903 (119905) 119861 (119905) 119889119905 119905 isin T 119861 (0) = 1 (4)

For each 119905 isin T and each 119896 = 1 2 suppose 120583119896(119905) 120590119896(119905) denote

the appreciation rate and the volatility rate of the 119896th riskyasset at time 119905 respectivelyThen the chain 120585 also determinesthe appreciation rate and volatility rate of the 119896th risky assetas

120583119896

(119905) = 120583119896

(119905 120585 (119905)) = ⟨120583119896 120585 (119905)⟩

120590119896

(119905) = 120590119896

(119905 120585 (119905)) = ⟨120590119896 120585 (119905)⟩

(5)

where 120583119896

= (1205831

119896 1205832

119896 120583

119873

119896)1015840

isin R119873 120590119896

= (1205901

119896 1205902

119896 120590

119873

119896)1015840

isin R119873 120583119894

119896gt 0 and 120590

119894

119896gt 0 for each 119894 = 1 2 119873 120583

119894

119896and 120590

119894

119896

are the appreciation rate and the volatility rate of the 119896th riskyasset at time 119905 respectively when the economy is in the 119894thstate at that time Furthermore we suppose that120583

119894

119896ge 119903119894for 119894 =

1 2 119873 and that 120583119894

119896rsquos and 120590

119894

119896rsquos are all distinctThe condition

120583119894

119896ge 119903119894is necessary to exclude the arbitrage opportunities in

the market


We consider two standard Brownian motions 1198821(119905) and

1198822(119905) for 119905 isin T on (ΩFP) To allow for complete gener-

ality we allow the two standard Brownian motions to becorrelated with correlation coefficient denoted by 120588 namely119864[1198821(119905)1198822(119905)] = 120588119905 In case there would only be one source

of randomness left in the model we also assume that 1205882

lt 1At the same time we also assume the two standard Brownianmotions are independent of the Markov chain 120585 For each119896 = 1 2 let 119878

119896= 119878119896(119905) | 119905 isin T denote the price process

of the 119896th risky asset Then we assume that the evolution of119878119896over time satisfies the following Markov regime switching

geometric Brownian motion

119889119878119896

(119905) = 120583119896

(119905) 119878119896

(119905) 119889119905 + 120590119896

(119905) 119878119896

(119905) 119889119882119896

(119905)

119878119896

(0) = 1199040

(6)

where the market price of risk for risky asset 119896 is defined as120579119896(119905) = (120583

119896(119905) minus 119903(119905))120590

119896(119905)

We consider stochastic dynamic portfolio game in acontinuous-time financial market between two investors callthem A and B Without loss of generality we suppose thatthere is a bank account that is freely available to both investorsand simultaneously there are only two correlated risky assetsin the financial market only one of which is available toeach investor Investor A may be allowed to trade in the firstrisky asset 119878

1 and similarly investor B may be restricted to

trade only in the second risky asset 1198782 They cooperate with

each other on investment by the choice of their individualdynamic portfolio trading strategies in the risky assets andbank account

In the next we describe the wealth dynamic processes ofboth investors For each 119905 isin T let 119883(119905 120587

1) denote the wealth

process of investor A at time 119905 under a portfolio strategy1205871

= 1205871(119905) | 119905 isin T with 119883(0) = 119909

0 Investor A invests

an amount of wealth 1205871(119905) in the risky asset 119878

1at time 119905 Note

that once1205871(119905) is determined the remaining amount invested

in the bank account is completely specified as 119883(119905 1205871)minus1205871(119905)

Similarly let 119884(119905 1205872) denote the wealth process of investor B

at time 119905 under a portfolio strategy 1205872

= 1205872(119905) | 119905 isin T with

119884(0) = 1199100 Investor B invests an amount of wealth 120587

2(119905) in the

risky asset 1198782at time 119905The remaining amount 119884(119905 120587

2)minus1205872(119905)

is in the bank accountLet Π

1be the space of all admissible portfolio strategies

1205871 The elements in Π

1satisfy the following two conditions

(i)F-progressivelymeasurable and cadlag (right-continuouswith left limit)R-valued process (ie 120587

1is a nonanticipative

function) and (ii) 119864[int119879

01205872

1(119905)119889119905] lt infin The condition (ii) is a

technical condition If 1205871

isin Π1 we call the portfolio strategy

1205871admissible So Π

1is the set of all admissible portfolio

strategies of investor A Similarly we can define the set of alladmissible portfolio strategies of investor B and denote it byΠ2As in a standard portfolio selection problem the portfolio

strategies (controls) are assumed to be piecewise continuousWe also assume the portfolio strategies of stochastic differen-tial game between the two investors are feedback strategiesmore specifically Markov control strategies Markov controlis only dependent on the current value of state variables in the

system not upon the history That is the value we choose attime 119905 only depends on the state of the system at this timeFurthermore the investor can condition his action at eachpoint in time on the basis of the state of the system at thatpoint in time In many cases it suffices to consider Markovcontrol For more discussions on the strategies employed inthe differential games interested readers can refer to [34]

We place no other restrictions on portfolio strategy 1205871

or 1205872 For example we allow 120587

1(119905) lt 0 or (120587

2(119905) lt 0) this

means the investors are allowed to sell the risky assets shortWhereas we allow 120587

1(119905) gt 119883(119905 120587

1) or (120587

2(119905) gt 119883(119905 120587

2))

this corresponds to a credit and it means the investors haveborrowed to purchase the risky assets Here we note thatthe investor decides the wealth amount allocated to therisky asset according to the current and past market pricesinformation and observations of market or macroeconomicconditions This is totally different from some traditionaloptimal portfolio models where the investors only considerthe price information in making their optimal investmentdecisions

Under the self-financing assumption for each 119905 isin T thedynamics of the wealth process 119883(119905 120587

1) associated with 120587

1of

investor A evolves over time as the following Markov regimeswitching stochastic differential equation

119889119883 (119905 1205871)

= [119903 (119905 120585 (119905)) 119883 (119905 1205871) + (120583

1(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

1(119905)] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905)

(7)

Similarly for each 119905 isin T the dynamics of the wealthprocess 119884(119905 120587


2of investor B is governed

by the following Markov-modulated stochastic differentialequation

119889119884 (119905 1205872)

= [119903 (119905 120585 (119905)) 119884 (119905 1205872) + (120583

2(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

2(119905)] 119889119905

+ 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(8)

For each 119905 isin T denote the sum of wealth processes by119885(12058711205872)(119905) = 119883(119905 120587

1) + 119884(119905 120587

2) Since 119883(119905 120587

1) and 119884(119905 120587

2)

are diffusion processes controlled by investors A and Brespectively then 119885

(12058711205872)(119905) is a jointly controlled diffusion

process Specifically the evolution of the sum process overtime is governed by the following Markov regime switchingstochastic differential equation

119889119885(12058711205872)

(119905)

= [119903 (119905 120585 (119905)) 119885(12058711205872)

(119905) + (1205831

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205871

(119905)

+ (1205832

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205872

(119905) ] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(9)

where 119885(12058711205872)(0) = 119909

0+ 1199100


For mathematical convenience we can rewrite (9) in amore compact form

119889119885(12058711205872)

(119905) = [119903 (119905) 119885(12058711205872)

(119905) + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905) ] 119889119905

+ 1205901

(119905) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905) 1205872

(119905) 1198891198822

(119905)

(10)

In the next section we provide a utility-based stochas-tic differential portfolio game with respect to the process119885(12058711205872) 119905 isin T of (9) or (10) We formulate the stochastic

differential portfolio game as a problem of maximizing theexpected utility of the sumof terminalwealth processesMoregeneral results on zero-sum stochastic differential portfoliogames are discussed in for example Browne [20] Moreoverfor some results on nonzero-sum differential games inter-ested readers can refer to Lin [29]

3 Nonzero-Sum Game Problem Formulation

In this section we consider nonzero-sum stochastic differ-ential portfolio game problem between two investors Thedifferential game is formulated as a problem to maximizingexpected utility of the sum of terminal wealth processes oftwo investors respectively at some fixed time 119879 isin T

For each 119896 = 1 2 let 119880119896

R+ rarr R denoteutility functions of investors A and B respectively whichare both strictly increasing strictly concave and continuousdifferentiable (ie 119880

1015840

119896gt 0 and 119880

10158401015840

119896lt 0) More results

about risk preference can refer to [35ndash37] Furthermore weassume that the utility functions satisfy the following Inadaconditions (technical conditions)

1198801015840

119896(0+) = lim

119911rarr0+

1198801015840

119896(119911) = +infin

1198801015840

119896(+infin) = lim

119911rarr+infin

1198801015840

119896(119911) = 0

(11)

In the case of two investors A and B for each 119905 isin T andeach 119894 = 1 2 119873 a typical differential game is posed asfollows Given 120585(119905) = e

119894and 119885

(12058711205872)(119905) = 119911 investor A choose

his own admissible portfolio strategy 1205871

isin Π1to maximize

119881(12058711205872)

1(119905 119911 e

119894)

= 119864 [1198801

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(12)

while investor B choose his own admissible portfolio strategy1205872

isin Π2to maximize

119881(12058711205872)

2(119905 119911 e

119894)

= 119864 [1198802

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(13)

with both utility maximization problems subject to the sumprocess (9) or (10)

We assume that each investor is aware of the otherinvestorrsquos presence and how the otherrsquos choice of his strategy

affects the state equation Furthermore we assume that thetwo investors choose their portfolio strategies simultaneouslyInvestor A would like to choose an admissible strategy 120587

1so

as to maximize his payoff 119881(12058711205872)

1(119905 119911 e

119894) for every possible

choice of investor Brsquos portfolio strategy while investor Bis trying to choose an admissible strategy 120587

2in order to

maximize his payoff 119881(12058711205872)

2(119905 119911 e

119894) for every possible choice

of investor Arsquos portfolio strategy The game terminates at afixed duration 119879 Then the nonzero-sum stochastic differen-tial portfolio game can be formulated as the following twooptimal portfolio selection utility maximization problems ofinvestors A and B

1198811

(119905 119911 e119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

1198812

(119905 119911 e119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(14)

Here 1198811(119905 119911 e

119894) and 119881

2(119905 119911 e

119894) are the value functions of

the optimal portfolio selection problems associated withinvestors A and B respectively over the time horizon [119905 119879]This is a two-player nonzero-sum stochastic differentialportfolio game between two investors A and B

To solve the nonzero-sum stochastic differential portfoliogame in the following we first give the definition of Nashequilibrium for the differential game between two investorsA and B described above

Definition 1 For each time 119905 isin T given that the state ofmacroeconomic is in the 119894th state let 120587

2be an admissible

strategy of investor B One defines the set of best responsesof investor A to the admissible portfolio strategy 120587

2as

119861119877119894

1(1205872)

= 120587lowast

1isin Π1

| 119881(120587lowast

11205872)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(12058711205872)

1(119905 119911 e

119894)

(15)

And similarly one can define the set of the best responses ofinvestor B to the strategy 120587

1of investor A as

119861119877119894

2(1205871)

= 120587lowast

2isin Π2

| 119881(1205871120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(12058711205872)

2(119905 119911 e

119894)

(16)

A pair of admissible portfolio strategies (120587lowast1

120587lowast

2) is said to be

a Nash equilibrium (ie saddle point) for the nonzero-sumdifferential game with investors A and B strategies spaces Π

1


and Π2 and payoffs (12) and (13) when the economy (Markov

chain) is in state e119894if

120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)

Equivalently (120587lowast1

120587lowast

2) is said to be a Nash equilibrium if

119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast

2are referred to as investor Arsquos and investor

Brsquos respective equilibrium strategies If a Nash equilibriumexists then the value functions of the nonzero-sum differ-ential game can be obtained For more discussions on theimplications of Nash equilibrium interested readers can referto [38]

4 Regime Switching HJB Equation andthe Optimal Conditions

In this section we will derive a pair of regime switching HJBequations and Feynman-Kac representations for the valuefunctions of the nonzero-sum differential game formulatedin the last section We will also derive a set of coupledHJB equations corresponding to the regime switching HJBequations

In the sequel we consider the case of investors withrisk averse exponential utility functions Suppose that utilityfunctions 119880

1of investor A and 119880

2of investor B are given by

1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574

2are positive constants which represent the

coefficients of absolute risk aversion (CARA) of investorsThat is

120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840

119896(119909) representing the first and second

derivatives of 119880119896with respect to 119909 for each 119896 = 1 2

Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each

119896 = 1 2 and each 119894 = 1 2 119873 We now solve the twoutilitymaximization problems via the dynamic programmingprinciple in stochastic optimal control according to [39] Itcan be shown that value functionsV

119896of utility maximization

of the nonzero-sum differential game satisfy the followingregime switching HJB equations

120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)

In what follows of this section with a slight abuse ofnotations for each 119896 = 1 2 we still let all the notations119903(119905 e119894) 120583119896(119905 e119894) 120590119896(119905 e119894) and 120587

119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587

119896(119905) for 119905 isin T unless otherwise stated

For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the

vector V119896of value functions at different regimes satisfies

the following two systems of coupled regime switching HJBequations respectively

120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)


with the terminal condition 1198811(119879 119911 e

119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911

In what follows to abbreviate the expression in curlybrackets for each 119896 = 1 2 we define the operators

L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)

Then the HJB equations (22) and (23) can be simplified asfollows

120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)

First the first-order conditions for maximizing the quan-tity in the HJB equations (25) and (26) give optimal portfoliostrategies

120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)

Second assume that HJB equations (22) or (25) and (23)or (26) have smooth solutions with 119881

119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for

each 119896 = 1 2 The subscripts on 119881119894

119896119911and 119881

119894

119896119911119911denote the first

and second partial differentiation with respect to variable 119911respectively Following the approach in [3] then we considervalue functions are of the following trial solutions

119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)

where 119891(119905 e119894) isin C1 to be determined is a suitable positive

function with the boundary condition 119891(119879 e119894) = 1 for all e

119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin

119864Hence from (27) we obtain the associated explicit expres-

sions of competitive optimal portfolio strategies

120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)


Last substituting these results into HJB equations (22) or(25) and (23) or (26) we derive the following two systems ofcoupled linear ordinary differential equations

120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)

It is well known that both systems of differential equationsonly have a unique solution respectively We will give anappropriate explanation in Remark 8 of this section Insummary we obtain the following theorem immediately

Theorem 2 If 1205882

= 1 then the value functions of investorsA and B at different regimes of the utility maximization ofnonzero-sum stochastic differential portfolio game are given by

119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e

119894) are solutions of coupled linear

ordinary differential equations (31) and (32) The optimalportfolio strategies can be expressed as

120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)

Remark 3 From the expressions (34) of optimal portfoliostrategies we can see that the optimal portfolio strategiesare composed of two terms and the first term is similar toMerton-type solution Meanwhile we find that the portfo-lio strategies also depend on the regime switching of themacroeconomy Both of them are directly proportional to theexpected excess returns on the corresponding risky assets inthe market when the correlation 120588 between two risky assetsis negative However when 120588 gt 0 the optimal strategiesare inversely proportional to the expected excess return ofthe other risky asset not available to the associated investorsFurthermore even if the macroeconomy has 119873 states only ifthe state of economy is unchanged we find that the optimalportfolio strategies are always constant Namely they areindependent of the wealth level of investors

Remark 4 When the Markov chain has only one state (ieS = s

1) then the Markov regime switching model con-

sidered here degenerates into a deterministic case And themarket parameters r120583 and120590 of themodel become constantsThen the conclusions will not be influenced byMarkov chainso we can omit the index 119894 in the value functions and theoptimal strategies Albeit with the difference in form thelinear ordinary differential equations which the functions 119891

and 119892 in the value functions satisfy coincide with (43) and(44) of lemma 41 of [29] respectively Correspondingly thevalue functions and optimal strategies are consistent with theresults of Theorem 41 of [29]

Corollary 5 If 120588 = 0 the two standard Brownian motions(1198821(119905) and119882

2(119905)) aremutual independent of (ΩFP)Then

the value functions of investors A and B at different regimesof the utility maximization of the nonzero-sum stochasticdifferential portfolio game are denoted by

119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)

and the optimal portfolio strategies are given by

120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)


where the functions 119891 and 119892 are the solutions of the followinglinear ordinary differential equations

120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)

Remark 6 If the two risky assets are not correlated witheach other from Corollary 5 we can see that the optimalinvestment strategies are of the Merton solution types at aparticular regime Moreover optimal investment strategiesare directly proportional to the expected return and inverselyproportional to the variance of the associated risky assetsavailable to investors at such a regime In this specialcase we know each investor does not want to considerthe other investorrsquos strategy when he makes his optimalportfolio choice It is equivalent to consider the classicalutility maximization for individualrsquos optimal portfolio choiceunder a regime switching financial market

We note that the functions 119891(119905 e119894) and 119892(119905 e

119894) in the

expressions of the value functions given inTheorem 2 satisfytwo systems of coupled linear ordinary differential equationsGenerally speaking differential equations are harder to solveIn what follows we will derive another representation of119891(119905 e119894) and 119892(119905 e

119894) via the Feynman-Kac type representations

which not only are more convenient to investigate the influ-ence of the Markov-modulation on the value functions of thenonzero-sum stochastic differential portfolio game problembut also can better interpret and derive some properties ofvalue functions To derive the Feynman-Kac representationsof both value functions 119891(119905 e

119894) and 119892(119905 e

119894) first we make the

following assumptions of notation

f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)

We now give the Feynman-Kac representations of 119891(119905 e119894)

and 119892(119905 e119894) in the following theorem

Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)

where diag [119909] denotes the diagonal matrix with diagonalelements given by the row vector 119909 Then the Feynman-Kacrepresentations of 119891(119905 e

119894) and 119892(119905 e

119894) are given by

119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e

119894) are denoted by (41) and (42) and 119864

119905119894

is the conditional expectation given that 120585(119905) = e119894underP

Proof First we will prove the Feynman-Kac representationof 119891(119905 e

119894) From Lemma 15 of Appendix B in [33] we know

that the process 119872(119904) | 119904 isin [119905 119879] defined by

119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)

is an (F120585P)-martingale Next we define a function 119877(119904) by

119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)


Then by applying the product rule of Ito formula and thedefinition of 119872(119904) we obtain

119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)

Setting 119904 = 119879 and taking expectation we obtain

119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)

We note that the proof of the Feynman-Kac representation offunction 119892(119905 e

119894) is the same as the above process so we omit

it here

From the expressions of 119891(119905 e119894) in (44) and 119892(119905 e

119894) in

(45) it is not difficult to see 119891(119905 e119894) gt 0 and 119892(119905 e

119894) gt 0

Consequently for each 119896 = 1 2 119881119896(119905 119911 e

119894) given by (33)

indeed satisfy 119881119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 just as described in the

previous section

Remark 8 By using the notations of (119860(119905) or 119861(119905)) and (f(119905)

or g(119905)) the matrix form of (31) and (32) can be denoted by

f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840

isin R119873 The differential equa-tion systems (50) indeed have unique continuous solutionsreferred to in (page 303Theorem 1) of [40] For more similardiscussions on the existence of solutions of the differentialequation system interested readers can refer to Remark 42of [41]

5 One Special Case

In order to investigate how the Markov regime switchinginfluences optimal investment strategies and value functionsand provide meaningful comparative statics analysis in thissection we present one special case of the nonzero-sumstochastic differential portfolio game problem established inSections 3 and 4Wewill assume that the interest rate denotedby 119903(119905 120585(119905)) equiv 119903 119905 isin T is not modulated by Markov chainNamely the price of bank account is not affected by theexternal conditions This assumption is relatively reasonablesince compared to risky assets the price of the bank accountis more stable Furthermore we assume S = 1 2 that is120585 = 120585(119905) | 119905 isin T is a two-state Markov chain We firstderive regime switching HJB equations for value functions ofthis simplifiedmodelWe assume that the ratematrix119876 of theMarkov chain is given by

119876 = (minus119902 119902

119902 minus119902) (51)

where 119902 is a positive real constant Note that it is not necessaryto consider the Markov chain with only two states It justincreases computation complexity in the numerical analysis

section given a general rate matrix 119876 Furthermore we knowthat the two-state Markov chain is rich enough to distinguisha ldquobullrdquo market and a ldquobearrdquo market In this case fromTheorem 2 we see that the value functions of investors Aand B for the two-state economy are denoted by the followingforms

119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)

where 119891(119905 119894) is the solutions of the following pair of linearordinary differential equations

120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)

with boundary conditions 119891(119879 1) = 1 and 119891(119879 2) = 1 and119892(119905 119894) is the solutions of the following pair of linear ordinarydifferential equations

120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)

with boundary conditions 119892(119879 1) = 1 and 119892(119879 2) = 1For the sake of conciseness we can rewrite (53)-(54) in

the following simplified forms by using the notations of 119886119894in

(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)

and the optimal portfolio strategies of investors A and B forthis case are respectively given by

120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)

We know that the above two pairs of linear differentialequations for functions 119891 and 119892 (see (55) and (56)) can beexplicitly solved To solve these two differential equations wefirst give the following lemma

Lemma9 Let 11990312

and 11990334

be the solutions of the following twoquadratic equations respectively

1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)

Then one can easily obtain

11990312

=minus (1198861

+ 1198862

minus 2119902) plusmn radic(1198861

minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)

where the notations 119886119894 119887119894 and 119902 are given by (41) (42) and

(51)

Proof The conclusion of this lemma follows directly fromthe solutions of quadratic equation Meanwhile the discrim-inants of the quadratic equations are always bigger than zeroso the equations have two different real roots

For mathematical convenience let us denote

1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)

Then the solutions of 119891(119905 1) 119891(119905 2) and 119892(119905 1) 119892(119905 2) areexplicitly given by

119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)

Remark 10 When the model parameters 120583119896and 120590

119896of risky

assets for each 119896 = 1 2 in the two regimes are identical toeach other respectively this result will lead to 119886

1= 1198862

= 119886

and 1198871

= 1198872

= 119887 No matter what the value of the parameter119902 in the rate matrix of the Markov chain is we find that theoptimal value functions and optimal portfolio strategies arerobust with respect to the change in the value of 119902 In this casethe above two pairs of linear ordinary differential equations(53) or (55) and (54) or (56)with the corresponding boundaryconditions have unique solutions respectively In this casewe can easily obtain the solutions of 119891 and 119892 That is

119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)


where this result is just as described in Remark 4 in Section 4This case is equivalent to the assumption that Markov chainhas only one state

6 Comparative Statics andNumerical Simulation

To gainmore insight into the economic significance of regimeswitching on the optimal portfolio strategies in this sectionwe will construct numerical analysis to investigate howoptimal investment strategies change with the parametersarising from our Markov-modulated model for example theabsolute risk aversion coefficients of two investors and thecorrelation coefficient between the two risky assets We alsomake comparisons of the qualitative behaviors of the optimalportfolio strategies obtained from our model (Model I) tothose arising from the model without regimes (Model II)

To perform both the comparative statics analysis and thecomparison between Model I and Model II simultaneouslywe implement the procedure just as [14] For illustrationwe consider the special case described in Section 5 whereMarkov chain is assumed to only have two states and interestrate is fixed Here we also suppose that the two states inMarkov chain 120585 represent economy 1 (E1) and economy 2(E2) respectively

In order to make the comparison effectively our numer-ical results are based on the following annualized baselinehypothetical values for the model parameters unless other-wise stated the absolute risk aversion coefficients 120574

1= 04

1205742

= 02 the current time 119905 = 6 the mature horizon 119879 = 10the risk-free interest rate 119903 = 005 the drift rates of thetwo risky assets 120583

1

1= 008 120583

1

2= 01 and the volatility

rates of the two risky assets 1205901

1= 03 120590

1

2= 05 These

hypothetical parameter values are drawn from those used in[29] because we simultaneously want to make comparisonsof the parameter sensitivity analysis results obtained fromourmodel to the properties obtained in [29]

For each 119896 = 1 2 when the drift rates 1205832

119896and the

volatility rates 1205902

119896of the risky assets in E2 are the same as their

corresponding parameter values in E1 we say that Model IandModel II are identical to each otherThenumerical resultsin this case for the optimal portfolio strategies obtained fromModel I are identical to those arising fromModel II whateverthe value of the parameter 119902 in the rate matrix of Markovchain 119876 is Thus the results are robust with respect to thechange of 119902 Interested readers can refer to Remark 10 fordetails about this description

In the next we implement the numerical analysis for theoptimal investment strategies with respect to that particularparameter We will focus on how the optimal portfoliostrategies vary against model parameters when the economystate modeled by Markov chain changes namely how theregime shifts inmarket parameters 120583

2

119896and 1205902

119896for each 119896 = 1 2

affect portfolio strategies In the following two subsectionswe analyze this issue along two dimensions Firstly we inves-tigate the impact of regime shifts in drift rates 120583

2

119896 119896 = 1 2 on

optimal portfolio strategies against the absolute risk aversioncoefficients and correlation coefficient respectively Secondly

005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05

Figure 1 Optimal strategy 1205871against 120583

2for different 120588

005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05



we analyze the effects of regime switching in volatilities offinancial assets on the portfolio strategies against the sameparameters as those with respect to the analysis of the impactof regime shifts in drift rates

61 The Effect of Drift Rates The main purpose of this sub-section is to see the effects of the drift rates 120583

2

1and 120583

2

2of

the two risky assets in economy E2 on the optimal portfoliostrategies We suppose there are three cases for our modelparameters namely 120583

2

1gt 1205831

1or 1205832

2gt 1205831

2or the two conditions

1205832

1gt 1205831

1and 120583

2

2gt 1205831

2satisfying simultaneously When only

one of the three conditions satisfies then economy E1 is saidto be a ldquogoodrdquo economy relative to economy E2 Or else E1 isa ldquobadrdquo economy In this case Model I and Model II are saidto be different from each other However for each 119896 = 1 2when 120583

1

119896= 1205832

119896(ie E1 and E2 coincide) it means that the two

states are identical It is equivalent to say the model has noregime switching and Model I and Model II are identical toeach other

Figures 1 and 2 plot the current optimal strategies 1205871

against the drift rate 1205832and 120587

2against the drift rate 120583

1for

two different opposite sign particular values of correlationcoefficient 120588 respectively Both figures show the fundamentalproperties of optimal portfolio strategies in a benchmarkmodel in which there are no regimes in the financial markets


minus08 minus06 minus04 minus02 0 02 04 06 08

120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6

Figure 3 Optimal 1205871in E2 against 120588 for different 120583

2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3

Figure 5 1205871in E2 against 120574

1for 120588 = 05 and different 120583

2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8


1for 120588 = minus05 and different 120583

2

119896

From Figure 1 it can be easily seen that 1205871decreases as 120583

2

increases when 120588 gt 0 while it increases when 120588 lt 0 Thisproperty of portfolio strategy 120587

1similarly applies to 120587

2

Figures 3 and 4 depict the plots of the optimal investmentstrategies 120587

1and 120587

2against the correlation coefficient 120588 for

different values of 1205832

119896 respectively The plots compare the

two cases when there is no regime (12058322

= 1205831

2= 01 120583

2

1=

1205831

1= 008) and when the state changes to E2 We can see that

the regime switching in drift rates has a significant impacton the qualitative behavior of optimal portfolio strategiesAccording to (57) and (58) for each 119896 = 1 2 we can easilyknow that 120587

119896increases with 120583

119896 From Figure 3 we can see

that 1205871increases along with 120583

2

1 When 120588 lt 0 it substantially

increases as 1205832

2 However it significantly decreases as 120583

2

2

increases when 120588 gt 0 According to Theorem 51 in [29] theoptimal portfolio strategy 120587

1decreases with 120588 when 120579

11205741

lt

12057921205741 Here the parameters given in Model I and Model II

satisfy this condition Likewise from Figure 4 we see that theoptimal investment 120587

2substantially increases along with 120583

2

2

and also increases as 1205832

1when 120588 lt 0 While 120588 gt 0 120587

2slightly

decreases as 1205832

1increases Moreover for the three cases when

1205832

2= 01 120583

2

2= 02 and 120583

2

1= 01 there is a critical point at

which there is a reversal of the behavior of optimal investmentstrategy fromdecreasing to increasing against different valuesof 120588

Figures 5 and 6 depict the plots of optimal investmentstrategy 120587

1against the absolute risk aversion coefficient 120574

1

Similar to the above in these figureswe compare the two caseswhen there is no regime (1205832

2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and

when the state changes to E2 From Figures 5 and 6 for allthe three cases when 120583

1

1= 008 120583

2

1= 01 and 120583

2

2= 03

we see that the optimal portfolio strategy 1205871decreases as

risk aversion coefficient 1205741increases Also 120587

1increases along

with 1205832

1whenever the values of correlation coefficient 120588 take

At the same time Figure 5 shows that optimal strategy 1205871

significantly decreases as 1205832

2increases when 120588 gt 0 However

we find that this result is opposite when 120588 lt 0 in Figure 6


02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896

Figures 7 and 8 plot 1205871against the absolute risk aversion

coefficient 1205742 From Figures 7 and 8 we see that the opti-

mal investment strategy increases as 1205832

1increases and also

substantially decreases as 1205832

2increases when 120588 gt 0 While

120588 lt 0 it shows the opposite result The optimal investmentsignificantly increases along with 120583

2

2 For all the three cases

when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2

when 120588 gt 0 When 120588 lt 0 it decreases with 1205742 This result is

consistent withTheorem 52 in [29]Figures 9 and 10 and Figures 11 and 12 describe the effects

of 1205832

119896on the qualitative behavior of the optimal portfolio

strategy 1205872of investor B against the risk aversion coefficients

1205741and 120574

2for different values of 120588 Since the explanations

for these figures are similar to the above analysis of optimalportfolio strategy of investor A we omit them here From theabove figures we can see that regime switching representedby the switches of drift rate120583 really has a significant impact onthe optimal investment strategy To consider regime switch-ingmodelswill help to incorporate the structure change of themodel and help investors to adjust their investment strategiesaccording to the changing conditions of the macroeconomy

62 The Effect of Volatility Rates In this subsection we willfocus on the effect of the volatility rates 120590

2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896

assets in E2 on the optimal portfolio strategies We supposethere are three cases for our model parameters namely 120590

2

1gt

1205901

1or 1205902

2gt 1205901

2or the two conditions 120590

2

1gt 1205901

1and 120590

2

2gt 1205901

2

satisfying simultaneously When one of the three conditionssatisfies then economy E1 is said to be a ldquobadrdquo economyrelative to E2 Or else E1 is a ldquogoodrdquo economy In this caseModel I and Model II are different from each other For each119896 = 1 2 when 120590

1

119896= 1205902

119896(ie E1 and E2 coincide) it means that

themodel has no regime and we sayModel I andModel II areidentical From the following comparative static analysis wecan see that the effects of regime switching on the optimalinvestment strategies are significant

Figures 13 and 14 describe the plots of optimal investmentstrategies 120587

1and 120587

2against the correlation coefficient for


119896 The plots compare when there is

no regime (12059022

= 1205901

2 1205902

1= 1205901

1) and when the state

changes to E2 From Figure 13 we see that the optimalinvestment 120587

1decreases when 120590

2

2increases for 120588 lt 0 This

result is opposite for 120588 gt 0 From Figure 14 it showsthat the optimal investment 120587


2

1increases

for 120588 lt 0 However for 120588 gt 0 it is an increasingfunction of 120590

2

1 This relation for both investment strategies

at a particular regime is consistent with Theorem 53 in [29]


02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896

And the monotonicity properties between two investmentstrategies and correlation coefficient for regime shifts involatilities are the same as the analysis of drift rates in Figures3 and 4

Figures 15 16 17 and 18 depict the plots of the optimalportfolio strategy 120587

1against the risk aversion coefficients 120574

1

and 1205742for different values of 120588 when the regime switching is

represented by switches of volatilities 1205902

119896 The results showed

in all these figures reflect that the regime switching involatility has a significant impact on the optimal portfoliostrategy The monotonicity properties between investmentand risk aversion coefficient 120574

1are the same for different

values of 120588 as what are described in detail in the abovesubsections for the effect of drift rates From Figure 15 when120588 takes the positive value and the volatility 120590

2

1increases from

03 to 05 we can see there is a critical point at which there is areversal of the optimal investment strategy from less to moreHowever in Figure 16 when 120588 = minus05 the optimal investmentstrategy in no regime (1205902

1= 03) is more than the case when

there is regime Based on the case for 1205902

1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1

Figure 13 1205871in E2 against 120588 for different 120590

2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896

shows that the optimal investment strategy becomes biggerwhen 120588 = 05 however this result is opposite in Figure 16

In Figures 17 and 18 we can see the monotonicityproperties of the optimal investment strategy 120587

1against the

risk aversion 1205742are the exact opposite when 120588 takes arbitrary

positive or negative value And also compared to the noregime switching case when 120590

2

1= 03 the optimal investment

strategy 1205871in the two regime switching cases (1205902

1= 05 or

1205902

1= 05 and 120590

2

2= 07) decreases significantly

Figures 19 20 21 and 22 plot the optimal investmentstrategy 120587


1and

1205742for different values of 120588 when the regime switching is

represented by switches of 1205902

119896 In Figures 19 and 20 we can

see the monotonicity relations between optimal investmentstrategies 120587

2and 1205741are exactly opposite when 120588 takes positive

or negative value while in Figures 21 and 22 the optimalinvestment strategy 120587

2decreases as 120574

2increases for different

values of 120588 Since the analysis for these figures is similar tothe above analysis of optimal investment strategy of investorA we omit it here


02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion

In this paper we dealt with a nonzero-sum stochastic dif-ferential portfolio game problem between two investors ina continuous-time Markov regime switching model Weformulated the stochastic differential game as twoutilitymax-imization problems We derived a pair of regime switchingHJB equations for this differential game problem and thenobtained two systems of coupled regime switching HJB equa-tions at different regimes Explicit solutions to the optimalinvestment strategies of two investors were also obtainedFurthermore we derived the Feynman-Kac representationsof the value functions of the two utility maximization prob-lems Numerical results for model parameters and the impactof the regime switching were illustrated and discussed whenthe Markov chain was assumed to only have two states Itshowed that the regime switching in the model parametershad a significant impact on the optimal portfolio strategies

There are some interesting and potential topics for futureresearch Firstly the present paper assumes that the Markov

chain can be observed However it is interesting to considera nonzero-sum stochastic differential game under a hiddenMarkovian regime switching economy Secondly we couldformulate the nonzero-sum stochastic differential game as amean-variance portfolio selection problem under a regimeswitching economy

A recent related publication by Bensoussan et al (2014)[42] studied noncooperative nonzero sum games and theproblem is formulated as utility maximization of the differ-ence between the terminal wealth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research for this paper was supported byThe Key Projectof National Natural Science Fund of China (Project no71431008) National Natural Science Foundation of China(Project no 11271375) National Natural Science InnovationResearch Group of China (Project no 71221001) and HunanProvincial Innovation Foundation For Postgraduate (Projectno CX2014B134)

References

[1] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol7 no 1 pp 77ndash91 1952

[2] R C Merton ldquoLifetime porfolio selection under uncertaintythe continuous-time caserdquoThe Review of Economics and Statis-tics vol 51 no 3 pp 247ndash257 1969

[3] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[4] R E Quandt ldquoThe estimation of the parameters of a linearregression system obeying two separate regimesrdquo Journal of theAmerican Statistical Association vol 53 no 284 pp 873ndash8801958

[5] J D Hamilton ldquoA new approach to the economic analysis ofnonstationary time series and the business cyclerdquo Econometricavol 57 no 2 pp 357ndash384 1989

[6] X Guo J Miao and E Morellec ldquoIrreversible investment withregime shiftsrdquo Journal of Economic Theory vol 122 no 1 pp37ndash59 2005

[7] H Chen ldquoMacroeconomic conditions and the puzzles of creditspreads and capital structurerdquo Journal of Finance vol 65 no 6pp 2171ndash2212 2010

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[9] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journal ofInformation Technology and Decision Making vol 8 no 4 pp787ndash801 2009

[10] N Bauerle and U Rieder ldquoPortfolio optimization withMarkov-modulated stock prices and interest ratesrdquo IEEE Transactions onAutomatic Control vol 49 no 3 pp 442ndash447 2004


[11] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 no 1 pp 271ndash291 2010

[12] R J Elliott T K Siu and A Badescu ldquoOn mean-varianceportfolio selection under a hiddenMarkovian regime-switchingmodelrdquo Economic Modelling vol 27 no 3 pp 678ndash686 2010

[13] T Honda ldquoOptimal portfolio choice for unobservable andregime-switchingmean returnsrdquo Journal of Economic Dynamicsand Control vol 28 no 1 pp 45ndash78 2003

[14] K C Yiu J Liu T K Siu and W-K Ching ldquoOptimalportfolios with regime switching and value-at-risk constraintrdquoAutomatica vol 46 no 6 pp 979ndash989 2010

[15] X Zhang and T K Siu ldquoOptimal investment and reinsuranceof an insurer with model uncertaintyrdquo Insurance Mathematicsamp Economics vol 45 no 1 pp 81ndash88 2009

[16] X Y Zhou and G Yin ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching a continuous-time modelrdquoSIAM Journal on Control and Optimization vol 42 no 4 pp1466ndash1482 2003

[17] R J Elliott ldquoThe existence of value in stochastic differentialgamesrdquo SIAM Journal on Control and Optimization vol 14 no1 pp 85ndash94 1976

[18] W H Fleming and P E Souganidis ldquoOn the existence ofvalue functions of two-player zero-sum stochastic differentialgamesrdquo Indiana University Mathematics Journal vol 38 no 2pp 293ndash314 1989

[19] D W K Yeung ldquoA stochastic differential game of institutionalinvestor speculationrdquo Journal of OptimizationTheory andAppli-cations vol 102 no 3 pp 463ndash477 1999

[20] S Browne ldquoStochastic differential portfolio gamesrdquo Journal ofApplied Probability vol 37 no 1 pp 126ndash147 2000

[21] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies amp Research vol 3 no 48 pp 235ndash254 2014

[22] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 439ndash4442013

[23] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[24] D W Yeung and L A Petrosyan Cooperative Stochastic Dif-ferential Games Springer Series in Operations Research andFinancial Engineering Springer New York NY USA 2006

[25] P Mannucci ldquoNonzero-sum stochastic differential games withdiscontinuous feedbackrdquo SIAM Journal on Control and Opti-mization vol 43 no 4 pp 1222ndash1233 2004

[26] R J Elliott and T K Siu ldquoA Markovian regime-switchingstochastic differential game for portfolio risk minimizationrdquo inProceedings of the American Control Conference (ACC rsquo08) pp1017ndash1022 Seattle Wash USA June 2008

[27] T K Siu ldquoA game theoretic approach to option valuation underMarkovian regime-switching modelsrdquo Insurance Mathematicsand Economics vol 42 no 3 pp 1146ndash1158 2008

[28] C K Leong and W Huang ldquoA stochastic differential game ofcapitalismrdquo Journal of Mathematical Economics vol 46 no 4pp 552ndash561 2010

[29] X Lin ldquoNonzero-sum stochastic differential portfolio gamesrdquoWorking Paper Zhenjiang Gongshang University 2013

[30] R J Elliott and T K Siu ldquoA stochastic differential gamefor optimal investment of an insurer with regime switchingrdquoQuantitative Finance vol 11 no 3 pp 365ndash380 2011

[31] J Z Liu and K F C Yiu ldquoOptimal stochastic differential gameswith VaR constraintsrdquo Discrete and Continuous DynamicalSystems Series B vol 18 no 7 pp 1889ndash1907 2013

[32] H Yang and L Zhang ldquoOptimal investment for insurer withjump-diffusion risk processrdquo Insurance Mathematics amp Eco-nomics vol 37 no 3 pp 615ndash634 2005

[33] R J Elliott L Aggoun and J Moore Hidden Markov ModelsEstimation and Control Springer New York NY USA 1995

[34] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement North Holland Amsterdam The Netherlands2nd edition 1991

[35] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[36] F Wen Z He X Gong and A Liu ldquoInvestorsrsquo risk preferencecharacteristics based on different reference pointrdquo DiscreteDynamics in Nature and Society vol 2014 Article ID 1583869 pages 2014

[37] C Huang H Kuang X Chen and F Wen ldquoAn LMI approachfor dynamics of switched cellular neural networks with mixeddelaysrdquo Abstract and Applied Analysis vol 2013 Article ID870486 8 pages 2013

[38] G L Esparza M G Torres and S L Torres ldquoA brief introduc-tion to differential gamesrdquo International Journal of Physical andMathematical Sciences vol 4 no 1 2013

[39] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 2006

[40] R BronsonMatrix Methods An Introduction Academic PressLondon UK 1991

[41] X Zhang and T K Siu ldquoOn optimal proportional reinsuranceand investment in a Markovian regime-switching economyrdquoActa Mathematica Sinica vol 28 no 1 pp 67ndash82 2012

[42] A Bensoussan C C Siu S C Yam and H Yang ldquoA class ofnon-zero-sum stochastic differential investment and reinsur-ance gamesrdquo Automatica vol 50 no 8 pp 2025ndash2037 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


random times References [7ndash9] established a dynamic capitalstructure model and demonstrated how business-cycle varia-tions (regime switching) in expected growth rates economicuncertainty and risk premia influence firmsrsquo financing anddefault policies Some papers on optimal portfolio selectionunder regime switching models can refer to [10ndash16] andothers However it seems that one important issue that maybe overlooked by the existing literature on portfolio selectionis interactive decision making problems In this paper wedevelop amodel of two investors to study interactive decisionmaking on portfolio selection

The origin of the differential game theory might trace tothe 1940s In 1965 Isaacswrote the classical work ldquoDifferentialGamerdquo It has laid down the solid mathematical and theoret-ical foundations to the differential game theory Since thenstochastic differential game models have found a wide rangeof applications in finance Some early works include [17ndash19] and some references therein Some recent works include[20ndash24] Browne [20] formulated various versions of a zero-sum stochastic differential game to investigate dynamicoptimal investment problems between two ldquosmallrdquo investorsin continuous time He provided the existence conditions ofNash equilibrium and gave explicit representations for thevalue functions and optimal portfolio strategies Mannucci[25] studied Nash equilibrium for two-player nonzero-sumstochastic differential game Elliott and Siu [26] extended themodel to a continuous-timeMarkovian regime switching set-ting and continued to study the risk minimization portfolioselection problem by using stochastic differential game Siu[27] considered option pricing under regime switching by agame theoretical approach More recently Leong and Huang[28] developed a stochastic differential game of capitalism tostudy the role of uncertainty Lin [29] studied a nonzero-sumstochastic differential portfolio game between two investorsElliott and Siu [30] introduced a model which coveredeconomic risk financial risk insurance risk and model riskto discuss an optimal investment problem of an insurancecompany using stochastic differential game approach Liuand Yiu [31] considered stochastic differential games withVaR risk constraints between two insurance companiesTheyprovided explicit Nash equilibria and derived closed-formsolutions to value functions However It seems that the lit-erature has not well studied the optimal portfolio interactivedecision making problem under stochastic differential gamein a continuous-time Markovian regime switching setting

To the best of our knowledge our model is most relatedwith Browne [20] and Lin [29] since the two papers havediscussed stochastic differential portfolio games Howeverthe distinction between them and our paper is also evidentThere are two fundamental differences between the modelconsidered by them and the models considered here Firstlywe incorporate our model with a general continuous-timeMarkov regime switching setting and take into accountthe Markov regime switching risk and its impacts on thefinancial asset prices Secondly we consider a nonzero-sumstochastic differential portfolio game which is different fromthe several versions of zero-sum stochastic differential gamein Browne [20] Furthermore we use a stochastic optimalcontrol approach for the current nonzero-sum stochastic

differential portfolio game problem This method is differentfrom the approach used in Lin [29] namely stochastic linearquadratic control

In this paper we treat the optimal portfolio selectionproblem in a wide class of continuous-time Markovianregime switchingmodelsWe consider the portfolio selectionbetween two ldquosmallrdquo investors call them A and B (Theinvestors are called ldquosmallrdquo as their decision making behaviordoes not affect the market prices of the underlying assets)We consider a continuous-time financial market with threeprimitive securities namely a bank account and two riskyassets The dynamic price processes of all the primitivesecurities are assumed to be modulated by a continuous-time Markovian chain The rationale of using this regimeswitching model is to incorporate the impact of regime shiftson asset prices attributed to structure changes in marketor macroeconomy The two risky assets are correlated witheach other only one of which is available to each investorMoreover both investors are allowed to trade freely in thebank accountThe investors cooperate with each other by thechoice of their own portfolio strategies when they make deci-sions on investment We formulate the stochastic differentialgame as two utility maximization problems Two objectivefunctions are considered here One investor is trying tomaximize his payoff simultaneously the other investor actsantagonistically to maximize the other payoff Each payoffis formulated as expected utility of the wealth sum processof the two investors By using stochastic optimal controltheory we derive a pair of regime switchingHJB equations forthe value functions Moreover we obtain the Feynman-Kacrepresentations of value functions Closed-form expressionsfor optimal portfolio strategies are also obtained Finally wefind that Markov regime switching in the model parametershas a significant effect on the optimal portfolio strategies andvalue functions

Aside from the intrinsic probabilistic and game theoreticinterest such a model is applicable in many economicsettings Aswe know diversification improves the ability of aninvestorrsquos risk-return trade-off However it can be difficult fora small investor to hold enough stocks which are well diver-sified In addition maintaining a well-diversified portfoliocan lead to high transaction costs If several investors forma group well-diversified portfolio and low transaction costscan be realized Different investors have different attitudestowards risk so the choice of the stock and the goal ofinvestment for investors are different with each other

The rest of the paper is organized as followsThe followingsection presents the price and wealth dynamic processes ina continuous-time Markov regime switching economy InSection 3 we first introduce two optimal portfolio problemswith different objective functions And then we formulatea two-investor nonzero-sum stochastic differential portfoliogame problem In Section 4 we derive a pair of regimeswitching Hamilton-Jacobi-Bellman (HJB) equations for thenonzero-sum differential game problem and explicit solu-tions for the optimal portfolio strategies and value functionsof two investors are obtained In Section 5 we discuss onespecial case for a two-state Markov regime switching modelIn Section 6weprovide the comparative statics andnumerical



2 Market Model



1199050le119905le119879





1 s2 s

119873 sub R119873



1 e2 e

119873 where e

119894isin R119873 and


119894119895







119894119895(119905) = P(120585(119905) = e

119895| 120585(0) = e

119894) for 119894 119895 =


= P(120585(0) = e119894)


]119894119895=12119873


119902119894119895

=

lim119905rarr0+

119875119894119895

(119905)

119905119894 = 119895

lim119905rarr0+

119875119894119894

(119905) minus 1

119905119894 = 119895

119894 119895 = 1 2 119873 (1)




119894119895ge 0 for 119894 = 119895 and sum

119873

119895=1119902119894119895

= 0so 119902119894119894





120585 (119905) = 120585 (0) + int119905

0

1198761015840120585 (119906) 119889119906 + M (119905) (2)




119903 (119905) = 119903 (119905 120585 (119905)) = ⟨r 120585 (119905)⟩ (3)


(1199031 1199032 119903

119873)1015840

isin R119873 with 119903119894




119889119861 (119905) = 119903 (119905) 119861 (119905) 119889119905 119905 isin T 119861 (0) = 1 (4)



120583119896

(119905) = 120583119896

(119905 120585 (119905)) = ⟨120583119896 120585 (119905)⟩

120590119896

(119905) = 120590119896

(119905 120585 (119905)) = ⟨120590119896 120585 (119905)⟩

(5)

where 120583119896

= (1205831

119896 1205832

119896 120583

119873

119896)1015840

isin R119873 120590119896

= (1205901

119896 1205902

119896 120590

119873

119896)1015840

isin R119873 120583119894

119896gt 0 and 120590

119894

119896gt 0 for each 119894 = 1 2 119873 120583

119894

119896and 120590

119894

119896


119894

119896ge 119903119894for 119894 =

1 2 119873 and that 120583119894

119896rsquos and 120590

119894


120583119894


the market










119889119878119896

(119905) = 120583119896

(119905) 119878119896

(119905) 119889119905 + 120590119896

(119905) 119878119896

(119905) 119889119882119896

(119905)

119878119896

(0) = 1199040

(6)


119896(119905) minus 119903(119905))120590

119896(119905)








= 1205871(119905) | 119905 isin T with 119883(0) = 119909








= 1205872(119905) | 119905 isin T with


2(119905) in the


2)minus1205872(119905)








01205872











1(119905) lt 0 or (120587

2(119905) lt 0) this


1(119905) gt 119883(119905 120587

1) or (120587

2(119905) gt 119883(119905 120587

2))




1of


119889119883 (119905 1205871)

= [119903 (119905 120585 (119905)) 119883 (119905 1205871) + (120583

1(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

1(119905)] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905)

(7)





119889119884 (119905 1205872)

= [119903 (119905 120585 (119905)) 119884 (119905 1205872) + (120583

2(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

2(119905)] 119889119905

+ 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(8)


1) + 119884(119905 120587

2) Since 119883(119905 120587

1) and 119884(119905 120587

2)




119889119885(12058711205872)

(119905)

= [119903 (119905 120585 (119905)) 119885(12058711205872)

(119905) + (1205831

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205871

(119905)

+ (1205832

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205872

(119905) ] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(9)

where 119885(12058711205872)(0) = 119909

0+ 1199100



119889119885(12058711205872)

(119905) = [119903 (119905) 119885(12058711205872)

(119905) + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905) ] 119889119905

+ 1205901

(119905) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905) 1205872

(119905) 1198891198822

(119905)

(10)





For each 119896 = 1 2 let 119880119896


1015840

119896gt 0 and 119880

10158401015840



1198801015840

119896(0+) = lim

119911rarr0+

1198801015840

119896(119911) = +infin

1198801015840


119911rarr+infin

1198801015840

119896(119911) = 0

(11)


119894and 119885



isin Π1to maximize

119881(12058711205872)

1(119905 119911 e

119894)

= 119864 [1198801

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(12)


isin Π2to maximize

119881(12058711205872)

2(119905 119911 e

119894)

= 119864 [1198802

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(13)




1so


1(119905 119911 e



2in order to


2(119905 119911 e



1198811

(119905 119911 e119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

1198812

(119905 119911 e119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(14)

Here 1198811(119905 119911 e

119894) and 119881

2(119905 119911 e





2be an admissible


2as

119861119877119894

1(1205872)

= 120587lowast

1isin Π1

| 119881(120587lowast

11205872)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(12058711205872)

1(119905 119911 e

119894)

(15)


1of investor A as

119861119877119894

2(1205871)

= 120587lowast

2isin Π2

| 119881(1205871120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(12058711205872)

2(119905 119911 e

119894)

(16)


120587lowast

2) is said to be


1




120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)


120587lowast


119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast








1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574



120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840



Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each




120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)


119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587


For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the



120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)



119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 Market Model



1199050le119905le119879





1 s2 s

119873 sub R119873



1 e2 e

119873 where e

119894isin R119873 and


119894119895







119894119895(119905) = P(120585(119905) = e

119895| 120585(0) = e

119894) for 119894 119895 =


= P(120585(0) = e119894)


]119894119895=12119873


119902119894119895

=

lim119905rarr0+

119875119894119895

(119905)

119905119894 = 119895

lim119905rarr0+

119875119894119894

(119905) minus 1

119905119894 = 119895

119894 119895 = 1 2 119873 (1)




119894119895ge 0 for 119894 = 119895 and sum

119873

119895=1119902119894119895

= 0so 119902119894119894





120585 (119905) = 120585 (0) + int119905

0

1198761015840120585 (119906) 119889119906 + M (119905) (2)




119903 (119905) = 119903 (119905 120585 (119905)) = ⟨r 120585 (119905)⟩ (3)


(1199031 1199032 119903

119873)1015840

isin R119873 with 119903119894




119889119861 (119905) = 119903 (119905) 119861 (119905) 119889119905 119905 isin T 119861 (0) = 1 (4)



120583119896

(119905) = 120583119896

(119905 120585 (119905)) = ⟨120583119896 120585 (119905)⟩

120590119896

(119905) = 120590119896

(119905 120585 (119905)) = ⟨120590119896 120585 (119905)⟩

(5)

where 120583119896

= (1205831

119896 1205832

119896 120583

119873

119896)1015840

isin R119873 120590119896

= (1205901

119896 1205902

119896 120590

119873

119896)1015840

isin R119873 120583119894

119896gt 0 and 120590

119894

119896gt 0 for each 119894 = 1 2 119873 120583

119894

119896and 120590

119894

119896


119894

119896ge 119903119894for 119894 =

1 2 119873 and that 120583119894

119896rsquos and 120590

119894


120583119894


the market










119889119878119896

(119905) = 120583119896

(119905) 119878119896

(119905) 119889119905 + 120590119896

(119905) 119878119896

(119905) 119889119882119896

(119905)

119878119896

(0) = 1199040

(6)


119896(119905) minus 119903(119905))120590

119896(119905)








= 1205871(119905) | 119905 isin T with 119883(0) = 119909








= 1205872(119905) | 119905 isin T with


2(119905) in the


2)minus1205872(119905)








01205872











1(119905) lt 0 or (120587

2(119905) lt 0) this


1(119905) gt 119883(119905 120587

1) or (120587

2(119905) gt 119883(119905 120587

2))




1of


119889119883 (119905 1205871)

= [119903 (119905 120585 (119905)) 119883 (119905 1205871) + (120583

1(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

1(119905)] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905)

(7)





119889119884 (119905 1205872)

= [119903 (119905 120585 (119905)) 119884 (119905 1205872) + (120583

2(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

2(119905)] 119889119905

+ 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(8)


1) + 119884(119905 120587

2) Since 119883(119905 120587

1) and 119884(119905 120587

2)




119889119885(12058711205872)

(119905)

= [119903 (119905 120585 (119905)) 119885(12058711205872)

(119905) + (1205831

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205871

(119905)

+ (1205832

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205872

(119905) ] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(9)

where 119885(12058711205872)(0) = 119909

0+ 1199100



119889119885(12058711205872)

(119905) = [119903 (119905) 119885(12058711205872)

(119905) + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905) ] 119889119905

+ 1205901

(119905) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905) 1205872

(119905) 1198891198822

(119905)

(10)





For each 119896 = 1 2 let 119880119896


1015840

119896gt 0 and 119880

10158401015840



1198801015840

119896(0+) = lim

119911rarr0+

1198801015840

119896(119911) = +infin

1198801015840


119911rarr+infin

1198801015840

119896(119911) = 0

(11)


119894and 119885



isin Π1to maximize

119881(12058711205872)

1(119905 119911 e

119894)

= 119864 [1198801

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(12)


isin Π2to maximize

119881(12058711205872)

2(119905 119911 e

119894)

= 119864 [1198802

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(13)




1so


1(119905 119911 e



2in order to


2(119905 119911 e



1198811

(119905 119911 e119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

1198812

(119905 119911 e119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(14)

Here 1198811(119905 119911 e

119894) and 119881

2(119905 119911 e





2be an admissible


2as

119861119877119894

1(1205872)

= 120587lowast

1isin Π1

| 119881(120587lowast

11205872)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(12058711205872)

1(119905 119911 e

119894)

(15)


1of investor A as

119861119877119894

2(1205871)

= 120587lowast

2isin Π2

| 119881(1205871120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(12058711205872)

2(119905 119911 e

119894)

(16)


120587lowast

2) is said to be


1




120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)


120587lowast


119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast








1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574



120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840



Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each




120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)


119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587


For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the



120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)



119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119889119878119896

(119905) = 120583119896

(119905) 119878119896

(119905) 119889119905 + 120590119896

(119905) 119878119896

(119905) 119889119882119896

(119905)

119878119896

(0) = 1199040

(6)


119896(119905) minus 119903(119905))120590

119896(119905)








= 1205871(119905) | 119905 isin T with 119883(0) = 119909








= 1205872(119905) | 119905 isin T with


2(119905) in the


2)minus1205872(119905)








01205872











1(119905) lt 0 or (120587

2(119905) lt 0) this


1(119905) gt 119883(119905 120587

1) or (120587

2(119905) gt 119883(119905 120587

2))




1of


119889119883 (119905 1205871)

= [119903 (119905 120585 (119905)) 119883 (119905 1205871) + (120583

1(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

1(119905)] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905)

(7)





119889119884 (119905 1205872)

= [119903 (119905 120585 (119905)) 119884 (119905 1205872) + (120583

2(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 120587

2(119905)] 119889119905

+ 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(8)


1) + 119884(119905 120587

2) Since 119883(119905 120587

1) and 119884(119905 120587

2)




119889119885(12058711205872)

(119905)

= [119903 (119905 120585 (119905)) 119885(12058711205872)

(119905) + (1205831

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205871

(119905)

+ (1205832

(119905 120585 (119905)) minus 119903 (119905 120585 (119905))) 1205872

(119905) ] 119889119905

+ 1205901

(119905 120585 (119905)) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905 120585 (119905)) 1205872

(119905) 1198891198822

(119905)

(9)

where 119885(12058711205872)(0) = 119909

0+ 1199100



119889119885(12058711205872)

(119905) = [119903 (119905) 119885(12058711205872)

(119905) + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905) ] 119889119905

+ 1205901

(119905) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905) 1205872

(119905) 1198891198822

(119905)

(10)





For each 119896 = 1 2 let 119880119896


1015840

119896gt 0 and 119880

10158401015840



1198801015840

119896(0+) = lim

119911rarr0+

1198801015840

119896(119911) = +infin

1198801015840


119911rarr+infin

1198801015840

119896(119911) = 0

(11)


119894and 119885



isin Π1to maximize

119881(12058711205872)

1(119905 119911 e

119894)

= 119864 [1198801

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(12)


isin Π2to maximize

119881(12058711205872)

2(119905 119911 e

119894)

= 119864 [1198802

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(13)




1so


1(119905 119911 e



2in order to


2(119905 119911 e



1198811

(119905 119911 e119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

1198812

(119905 119911 e119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(14)

Here 1198811(119905 119911 e

119894) and 119881

2(119905 119911 e





2be an admissible


2as

119861119877119894

1(1205872)

= 120587lowast

1isin Π1

| 119881(120587lowast

11205872)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(12058711205872)

1(119905 119911 e

119894)

(15)


1of investor A as

119861119877119894

2(1205871)

= 120587lowast

2isin Π2

| 119881(1205871120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(12058711205872)

2(119905 119911 e

119894)

(16)


120587lowast

2) is said to be


1




120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)


120587lowast


119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast








1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574



120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840



Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each




120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)


119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587


For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the



120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)



119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119885(12058711205872)

(119905) = [119903 (119905) 119885(12058711205872)

(119905) + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905) ] 119889119905

+ 1205901

(119905) 1205871

(119905) 1198891198821

(119905) + 1205902

(119905) 1205872

(119905) 1198891198822

(119905)

(10)





For each 119896 = 1 2 let 119880119896


1015840

119896gt 0 and 119880

10158401015840



1198801015840

119896(0+) = lim

119911rarr0+

1198801015840

119896(119911) = +infin

1198801015840


119911rarr+infin

1198801015840

119896(119911) = 0

(11)


119894and 119885



isin Π1to maximize

119881(12058711205872)

1(119905 119911 e

119894)

= 119864 [1198801

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(12)


isin Π2to maximize

119881(12058711205872)

2(119905 119911 e

119894)

= 119864 [1198802

[119885(12058711205872)

(119879)] | 119885(12058711205872)

(119905) = 119911 120585 (119905) = e119894]

(13)




1so


1(119905 119911 e



2in order to


2(119905 119911 e



1198811

(119905 119911 e119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

1198812

(119905 119911 e119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(14)

Here 1198811(119905 119911 e

119894) and 119881

2(119905 119911 e





2be an admissible


2as

119861119877119894

1(1205872)

= 120587lowast

1isin Π1

| 119881(120587lowast

11205872)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(12058711205872)

1(119905 119911 e

119894)

(15)


1of investor A as

119861119877119894

2(1205871)

= 120587lowast

2isin Π2

| 119881(1205871120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(12058711205872)

2(119905 119911 e

119894)

(16)


120587lowast

2) is said to be


1




120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)


120587lowast


119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast








1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574



120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840



Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each




120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)


119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587


For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the



120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)



119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120587lowast

1isin 119861119877119894

1(120587lowast

2) 120587

lowast

2isin 119861119877119894

2(120587lowast

1) (17)


120587lowast


119881(120587lowast

1120587lowast

2)

1(119905 119911 e

119894) = sup1205871isinΠ1

119881(1205871120587lowast

2)

1(119905 119911 e

119894)

119881(120587lowast

1120587lowast

2)

2(119905 119911 e

119894) = sup1205872isinΠ2

119881(120587lowast

11205872)

2(119905 119911 e

119894)

(18)

where 120587lowast

1and 120587

lowast








1198801

(119909) = minus1

1205741

119890minus1205741119909 119880

2(119909) = minus

1

1205742

119890minus1205742119909 (19)

where 1205741and 120574



120574119896

= minus11988010158401015840

119896(119909)

1198801015840119896

(119909) 119896 = 1 2 (20)

with 1198801015840

119896(119909) and 119880

10158401015840



Let 119881119894

119896= 119881119896(119905 119911 e

119894) and V

119896= (1198811

119896 1198812

119896 119881

119873

119896) for each




120597V1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597V1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972V1

1205971199112

+ ⟨V1 1198761015840120585 (119905)⟩ = 0

120597V2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597V2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972V2

1205971199112

+ ⟨V2 1198761015840120585 (119905)⟩ = 0

(21)


119896(119905 e119894) be denoted by 119903(119905)

120583119896(119905) 120590119896(119905) and 120587


For each 119894 = 1 2 119873 let 119881119894

119896= 119881119896(119905 119911 119890

119894) Hence the



120597119881119894

1

120597119905+ sup1205871isinΠ1

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905)

+ (1205832

(119905) minus 119903 (119905)) 120587lowast

2(119905)]

120597119881119894

1

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) (120587lowast

2(119905))2

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 120587lowast

2(119905) ]

1205972119881119894

1

1205971199112

+sum119895isin119864

119902119894119895

[1198811

(119905 119911 e119895) minus 1198811

(119905 119911 e119894)]

= 0

(22)



119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894) = 119880

1(119911) =

minus(11205741)119890minus1205741119911

120597119881119894

2

120597119905+ sup1205872isinΠ2

[119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 120587lowast

1(119905)

+ (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

2

120597119911

+1

2[1205902

1(119905) (120587lowast

1(119905))2

+ 1205902

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 120587lowast

1(119905) 1205872

(119905) ]1205972119881119894

2

1205971199112

+sum119895isin119864

119902119894119895

[1198812

(119905 119911 e119895) minus 1198812

(119905 119911 e119894)]

= 0

(23)


119894) = 119880

2(119911) =

minus(11205742)119890minus1205742119911


L(12058711205872)119881119896

(119905 119911 e119894)

= [119903 (119905) 119911 + (1205831

(119905) minus 119903 (119905)) 1205871

(119905) + (1205832

(119905) minus 119903 (119905)) 1205872

(119905)]120597119881119894

119896

120597119911

+1

2[1205902

1(119905) 1205872

1(119905) + 120590

2

2(119905) 1205872

2(119905)

+ 21205881205901

(119905) 1205902

(119905) 1205871

(119905) 1205872

(119905) ]1205972119881119894

119896

1205971199112

+ sum119895isin119864

119902119894119895

[119881119896

(119905 119911 e119895) minus 119881119896

(119905 119911 e119894)] 119894 = 1 2 119873

(24)


120597119881119894

1

120597119905+ sup1205871isinΠ1

L(1205871120587lowast

2)1198811

(119905 119911 e119894) = 0 (25)

120597119881119894

2

120597119905+ sup1205872isinΠ2

L(120587lowast

11205872)1198812

(119905 119911 e119894) = 0 (26)


120587lowast

1(119905) =

1

1 minus 1205882(120588

1205832

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

minus1205831

(119905) minus 119903 (119905)

12059021

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

)

120587lowast

2(119905) =

1

1 minus 1205882(120588

1205831

(119905) minus 119903 (119905)

1205901

(119905) 1205902

(119905)

120597119881119894

1120597119911

120597211988111989411205971199112

minus1205832

(119905) minus 119903 (119905)

12059022

(119905)

120597119881119894

2120597119911

120597211988111989421205971199112

)

(27)


119894

119896119911gt 0 and 119881

119894

119896119911119911lt 0 for


119896119911and 119881

119894



119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894) (28)



119894isin

119864 Consider

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894) (29)



119894isin



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(30)



120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119891 (119905 e119894)

120597119905+

1205742

1

2 (1 minus 1205882)

sdot [(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057421

1205901

(119905) 1205902

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(31)

120597119892 (119905 e119894)

120597119905+

1205742

2

2 (1 minus 1205882)

sdot [(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

12057422

1205901

(119905) 1205902

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 e

119894) = 1

(32)




119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(33)

where 119891(119905 e119894) and 119892(119905 e



120587lowast

1(119905) =

1

1 minus 1205882(

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205832

(119905) minus 119903 (119905)

12057421205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

120587lowast

2(119905) =

1

1 minus 1205882(

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

minus 1205881205831

(119905) minus 119903 (119905)

12057411205901

(119905) 1205902

(119905)119890minusint119879

119905119903(119904)119889119904

)

(34)









119881119894

1= 1198811

(119905 119911 e119894) = minus

1

1205741

119890minus1205741119911119890int119879

119905119903(119904)119889119904

119891 (119905 e119894)

119881119894

2= 1198812

(119905 119911 e119894) = minus

1

1205742

119890minus1205742119911119890int119879

119905119903(119904)119889119904

119892 (119905 e119894)

(35)


120587lowast

1(119905) =

1205831

(119905) minus 119903 (119905)

120574112059021

(119905)119890minusint119879

119905119903(119904)119889119904

120587lowast

2(119905) =

1205832

(119905) minus 119903 (119905)

120574212059022

(119905)119890minusint119879

119905119903(119904)119889119904

(36)



120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119891 (119905 e119894)

120597119905+

1205742

1

2[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)] 119891 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119891 (119905 e119895) minus 119891 (119905 e

119894)] = 0 119891 (119879 e

119894) = 1

(37)

120597119892 (119905 e119894)

120597119905+

1205742

2

2[

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)] 119892 (119905 e

119894)

+ sum119895isin119864

119902119894119895

[119892 (119905 e119895) minus 119892 (119905 e

119894)] = 0 119892 (119879 119894) = 1

(38)



119894) in the




119894) and 119892(119905 e



f (119905) = (119891 (119905 e1) 119891 (119905 e

2) 119891 (119905 e

119873))1015840

(39)

g (119905) = (119892 (119905 e1) 119892 (119905 e

2) 119892 (119905 e

119873))1015840

(40)

119886119894

= 119886 (119905 e119894)

=1205742

1

2 (1 minus 1205882)[

(1205832

(119905) minus 119903 (119905))2

12057422

12059022

(119905)minus

(1205831

(119905) minus 119903 (119905))2

1205742112059021

(119905)

minus 2(1205832

(119905) minus 119903 (119905))2

1205741120574212059022

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574211205901

(119905) 1205902

(119905)]

(41)

119887119894

= 119887 (119905 e119894)

=1205742

2

2 (1 minus 1205882)[

(1205831

(119905) minus 119903 (119905))2

12057421

12059021

(119905)minus

(1205832

(119905) minus 119903 (119905))2

1205742212059022

(119905)

minus 2(1205831

(119905) minus 119903 (119905))2

1205741120574212059021

(119905)

+ 2120588(1205831

(119905) minus 119903 (119905)) (1205832

(119905) minus 119903 (119905))

120574221205901

(119905) 1205902

(119905)]

(42)



Theorem 7 Let

119860 (119905) = diag [119886 (119905 e1) 119886 (119905 e

2) 119886 (119905 e

119873)] minus 119876

119861 (119905) = diag [119887 (119905 e1) 119887 (119905 e

2) 119887 (119905 e

119873)] minus 119876

(43)


119894) and 119892(119905 e


119891 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)] (44)

119892 (119905 e119894) = 119864119905119894

[exp(int119879

119905

119887 (119906 120585 (119906)) 119889119906)] (45)

where 119886(119905 e119894) and 119887(119905 e


119905119894





119872 (119904) = 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905))

minus int119904

119905

(119891119905

(119906 120585 (119906)) + ⟨f (119906) 1198761015840120585 (119906)⟩) 119889119906

(46)


119877 (119904) = exp(int119904

119905

119886 (119906 120585 (119906)) 119889119906) (47)



119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 (119904) 119891 (119904 120585 (119904)) minus 119891 (119905 120585 (119905)) = int119904

119905

119877 (119906) 119889119872 (119906) (48)


119891 (119905 e119894) = 119864119905119894

[119877 (119879)] = 119864119905119894

[exp(int119879

119905

119886 (119906 120585 (119906)) 119889119906)]

(49)



it here


119894) in


119894) gt 0


119894) given by (33)


119896119911gt 0 and 119881

119894


previous section



f1015840 (119905) = 119860 (119905) f (119905) f (119879) = 1

g1015840 (119905) = 119861 (119905) g (119905) g (119879) = 1

(50)

where 1 = (1 1 1)1015840


5 One Special Case


119876 = (minus119902 119902

119902 minus119902) (51)



119881119894

1= 1198811

(119905 119911 119894) = minus1

1205741

119890minus1205741119911119890119903(119879minus119905)

119891 (119905 119894) 119894 = 1 2

119881119894

2= 1198812

(119905 119911 119894) = minus1

1205742

119890minus1205742119911119890119903(119879minus119905)

119892 (119905 119894) 119894 = 1 2

(52)


120597119891 (119905 1)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205831

2minus 119903)2

12057422

(12059012)2

minus(1205831

1minus 119903)2

12057421

(12059011)2

minus 2(1205831

2minus 119903)2

12057411205742

(12059012)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

12057421

1205901112059012

]

]

119891 (119905 1)

+ 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

120597119891 (119905 2)

120597119905+

1205742

1

2 (1 minus 1205882)[

[

(1205832

2minus 119903)2

12057422

(12059022)2

minus(1205832

1minus 119903)2

12057421

(12059021)2

minus 2(1205832

2minus 119903)2

12057411205742

(12059022)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057421

1205902112059022

]

]

119891 (119905 2)

minus 119902 [119891 (119905 2) minus 119891 (119905 1)] = 0

(53)


120597119892 (119905 1)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205831

1minus 119903)2

12057421

(12059011)2

minus(1205831

2minus 119903)2

12057422

(12059012)2

minus 2(1205831

1minus 119903)2

12057411205742

(12059011)2

+ 2120588(1205831

1minus 119903) (120583

1

2minus 119903)

120574221205901112059012

]

]

119892 (119905 1)

+ 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0


120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120597119892 (119905 2)

120597119905+

1205742

2

2 (1 minus 1205882)[

[

(1205832

1minus 119903)2

12057421

(12059021)2

minus(1205832

2minus 119903)2

12057422

(12059022)2

minus 2(1205832

1minus 119903)2

12057411205742

(12059021)2

+ 2120588(1205832

1minus 119903) (120583

2

2minus 119903)

12057422

1205902112059022

]

]

119892 (119905 2)

minus 119902 [119892 (119905 2) minus 119892 (119905 1)] = 0

(54)



(41) and 119887119894in (42)

120597119891 (119905 1)

120597119905+ (1198861

minus 119902) 119891 (119905 1) + 119902119891 (119905 2) = 0 119891 (119879 1) = 1

120597119891 (119905 2)

120597119905+ (1198862

minus 119902) 119891 (119905 2) + 119902119891 (119905 1) = 0 119891 (119879 2) = 1

(55)

120597119892 (119905 1)

120597119905+ (1198871

minus 119902) 119892 (119905 1) + 119902119892 (119905 2) = 0 119892 (119879 1) = 1

120597119892 (119905 2)

120597119905+ (1198872

minus 119902) 119892 (119905 2) + 119902119892 (119905 1) = 0 119892 (119879 2) = 1

(56)


120587lowast

1(119905 1) =

1

1 minus 1205882(

1205831

1minus 119903

1205741

(12059011)2

119890minus119903(119879minus119905)

minus 1205881205831

2minus 119903

12057421205901112059012

119890minus119903(119879minus119905)

)

120587lowast

1(119905 2) =

1

1 minus 1205882(

1205832

1minus 119903

1205741

(12059021)2

119890minus119903(119879minus119905)

minus 1205881205832

2minus 119903

12057421205902112059022

119890minus119903(119879minus119905)

)

(57)

120587lowast

2(119905 1) =

1

1 minus 1205882(

1205831

2minus 119903

1205742

(12059012)2

119890minus119903(119879minus119905)

minus 1205881205831

1minus 119903

12057411205901112059012

119890minus119903(119879minus119905)

)

120587lowast

2(119905 2) =

1

1 minus 1205882(

1205832

2minus 119903

1205742

(12059022)2

119890minus119903(119879minus119905)

minus 1205881205832

1minus 119903

12057411205902112059022

119890minus119903(119879minus119905)

)

(58)


Lemma9 Let 11990312

and 11990334


1199032

+ (1198861

+ 1198862

minus 2119902) 119903 + [(1198861

minus 119902) (1198862

minus 119902) minus 1199022] = 0 (59)

1199032

+ (1198871

+ 1198872

minus 2119902) 119903 + [(1198871

minus 119902) (1198872

minus 119902) minus 1199022] = 0 (60)


11990312

=minus (1198861

+ 1198862


minus 1198862)2

+ 41199022

2

11990334

=minus (1198871

+ 1198872


minus 1198872)2

+ 41199022

2

(61)


(51)



1198621

=(1199032

+ 1198862) (1199031

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198622

= minus(1199031

+ 1198862) (1199032

+ 1198862

minus 119902)

119902 (1199031

minus 1199032)

1198623

= minus(1199032

+ 1198862)

(1199031

minus 1199032)

1198624

=(1199031

+ 1198862)

(1199031

minus 1199032)

1198625

=(1199034

+ 1198872) (1199033

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198626

= minus(1199033

+ 1198872) (1199034

+ 1198872

minus 119902)

119902 (1199033

minus 1199034)

1198627

= minus(1199034

+ 1198872)

(1199033

minus 1199034)

1198628

=(1199033

+ 1198872)

(1199033

minus 1199034)

(62)


119891 (119905 1) = 1198621119890minus1199031(119879minus119905)

+ 1198622119890minus1199032(119879minus119905)

119891 (119905 2) = 1198623119890minus1199031(119879minus119905)

+ 1198624119890minus1199032(119879minus119905)

119892 (119905 1) = 1198625119890minus1199033(119879minus119905)

+ 1198626119890minus1199034(119879minus119905)

119892 (119905 2) = 1198627119890minus1199033(119879minus119905)

+ 1198628119890minus1199034(119879minus119905)

(63)


119896of risky


1= 1198862

= 119886

and 1198871

= 1198872


119891 (119905) = 119890119886(119879minus119905)

119892 (119905) = 119890119887(119879minus119905)

(64)







1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1= 04

1205742


1

1= 008 120583

1



1= 03 120590

1

2= 05 These



119896and the





2

119896and 1205902

119896for each 119896 = 1 2


2

119896 119896 = 1 2 on


005 01 015 02 025 03

1205832

minus4

minus2

0

2

4

6

1205871

t = 6

120588 = 05

120588 = minus05



005 01 015 02 025 03

1205831

minus2

minus1

0

1

2

3

4

1205872

t = 6

120588 = 05

120588 = minus05





2

1and 120583

2

2of


2

1gt 1205831

1or 1205832

2gt 1205831


1205832

1gt 1205831

1and 120583

2

2gt 1205831



1

119896= 1205832






1for




120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588

12058321 = 008

12058321 = 01

12058322 = 03

minus15

minus10

minus5

0

5

10

15

20

12058712

t = 6


2

119896


120588

0

1

2

3

4

5

6

7

8

9

12058722

t = 6

12058322 = 01

12058322 = 02

12058321 = 01


2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus5

minus4

minus3

minus2

minus1

0

1

2

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7

8



2

119896


2



2


1and 120587





= 1205831

2= 01 120583

2

1=

1205831






2




2

2



11205741

lt




2

2


1when 120588 lt 0 While 120588 gt 0 120587

2slightly



1205832

2= 01 120583

2

2= 02 and 120583

2





1


2= 1205831

2= 01 120583

2

1= 1205831

1= 008) and


1

1= 008 120583

2

1= 01 and 120583

2

2= 03



1increases along

with 1205832







02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

t = 6 120588 = 05

12058712

minus4

minus3

minus2

minus1

0

1

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058321 = 008

12058321 = 01

12058322 = 03

12058712

t = 6 120588 = minus05

1

2

3

4

5

6

7



2

119896




1increases and also




2


when 1205831

2= 01 120583

2

2= 02 and 120583

2

1= 01 120587


2



of 1205832



1205741and 120574




2

1and 120590

2

2of the risky

02 03 04 05 06 07 08 09 1 11 12

1205741

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01

12058722

05

1

15

2

25

3

35

4



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12058322 = 01

12058322 = 02

12058321 = 01

12058722

t = 6 120588 = minus05

1

15

2

25

3

35

4

45

5



2

119896


2

1gt

1205901

1or 1205902

2gt 1205901


2

1gt 1205901

1and 120590

2

2gt 1205901

2


1

119896= 1205902




1and 120587




no regime (12059022

= 1205901

2 1205902

1= 1205901




2




2

1increases


2




02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 07 08 09 1 11 12

1205742

minus05

0

05

1

15

2

25

3

35

12058722

t = 6 120588 = 05

12058322 = 01

12058322 = 02

12058321 = 01



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12058722

12058322 = 01

12058322 = 02

12058321 = 01

05

1

15

2

25

3

35

4

45t = 6 120588 = minus05



2

119896




1







2

1increases from




1= 05 Figure 15


120588

0

1

2

3

4

5

12058712

t = 6

12059021 = 03

12059021 = 05

12059022 = 07

minus2

minus1


2

119896


120588

0

05

1

15

2

25

3

35

12058722

t = 6

12059022 = 05

12059022 = 07

12059021 = 05


2

119896



1against the



2



1= 05 or

1205902

1= 05 and 120590

2




1and











02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = 05

minus08

minus06

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

0

05

1

15

2

25

3



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

t = 6 120588 = 05

12058712

minus04

minus02

0

02

04

06

08

1



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059021 = 03

12059021 = 05

12059022 = 07

12058712

t = 6 120588 = minus05

02

04

06

08

1

12

14

16

18

2



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = 05

01

02

03

04

05

06

07

08

09

1

11



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205741

12059022 = 05

12059022 = 07

12059021 = 05

12058722

t = 6 120588 = minus05

04

06

08

1

12

14

16

18

2



2

119896


02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 07 08 09 1 11 12

1205742

t = 6 120588 = 05

12058722

12059022 = 05

12059022 = 07

12059021 = 05

minus02

0

02

04

06

08

1

12



2

119896

02 03 04 05 06 07 08 09 1 11 12

1205742

12059022 = 05

12059022 = 07

12059021 = 05

12058722

02

04

06

08

1

12

14

16t = 6 120588 = minus05



2

119896

7 Conclusion







Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Nonzero-Sum Stochastic Differential Portfolio Games...

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