Mean-field stochastic differential equations and ...

The Annals of Probability2017 Vol 45 No 2 824ndash878DOI 10121415-AOP1076copy Institute of Mathematical Statistics 2017

MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONSAND ASSOCIATED PDES

BY RAINER BUCKDAHNlowastDagger JUAN LIdagger1SHIGE PENGDagger AND CATHERINE RAINERlowast

Universiteacute de Bretagne Occidentalelowast Shandong University Weihaidagger

and Shandong UniversityDagger

In this paper we consider a mean-field stochastic differential equa-tion also called the McKeanndashVlasov equation with initial data (t x) isin[0 T ] times R

d whose coefficients depend on both the solution Xtxs and its

law By considering square integrable random variables ξ as initial condi-tion for this equation we can easily show the flow property of the solution

Xtξs of this new equation Associating it with a process X

txPξs which co-

incides with Xtξs when one substitutes ξ for x but which has the advan-

tage to depend on ξ only through its law Pξ we characterize the function

V (t xPξ ) = E[(XtxPξ

T PX

tξT

)] under appropriate regularity conditions

on the coefficients of the stochastic differential equation as the unique clas-sical solution of a nonlocal partial differential equation of mean-field typeinvolving the first- and the second-order derivatives of V with respect to itsspace variable and the probability law The proof bases heavily on a pre-liminary study of the first- and second-order derivatives of the solution ofthe mean-field stochastic differential equation with respect to the probabil-ity law and a corresponding Itocirc formula In our approach we use the notionof derivative with respect to a probability measure with finite second mo-ment introduced by Lions in [Cours au Collegravege de France Theacuteorie des jeu agravechamps moyens (2013)] and we extend it in a direct way to the second-orderderivatives

CONTENTS

1 Introduction 8252 Preliminaries 8283 The mean-field stochastic differential equation 8354 First-order derivatives of XtxPξ 8385 Second-order derivatives of XtxPξ 8536 Regularity of the value function 863

Received October 2014 revised October 20151Supported by the NSF of P R China (Nos 11071144 11171187 11222110) Shandong Province

(Nos BS2011SF010 JQ201202) Program for New Century Excellent Talents in University (NoNCET-12-0331) 111 Project (No B12023)

MSC2010 subject classifications Primary 60H10 secondary 60K35Key words and phrases Mean-field stochastic differential equation McKeanndashVlasov equation

value function PDE of mean-field type

824

MEAN-FIELD SDES AND ASSOCIATED PDES 825

7 Itocirc formula and PDE associated with mean-field SDE 870Acknowledgments 877References 877

1 Introduction Given a complete probability space (FP ) endowedwith a Brownian motion B = (Bt )tisin[0T ] and its filtration F = (Ft )tisin[0T ] aug-mented by all P -null sets and a sufficiently rich sub-σ -algebra F0 independentof B we consider the mean-field stochastic differential equation (SDE) alsoknown under the name of McKeanndashVlasov SDE

dXtxs = σ

(Xtx

s PXtxs

)dBs + b

(Xtx

s PXtxs

)ds

(11)s isin [t T ]Xtx

t = x isinRd

It is well known that under an appropriate Lipschitz assumption on the coeffi-cients this equation possesses for all (t x) isin [t T ]timesR

d a unique solution Xtxs s isin

[0 T ] For the classical SDE whose coefficients σ(xμ) = σ(x) b(xμ) = b(x)

depend only on x isin Rd but not on the probability measure μ it is well known

that the solution Xtxs 0 le t le s le T x isinR

d defines a flow and if the coefficientsare regular enough the unique classical solution of the partial differential equation(PDE)

parttV (t x) + 12 tr

(σσ lowast(x)D2

xV (t x)) + b(x)DxV (t x) = 0

(t x) isin [0 T ] timesRd(12)

V (T x) = (x) x isin Rd

is V (t x) = E[(XtxT )] (t x) isin [0 T ] times R

d But how about the above SDEwhose coefficients depend on (xμ) isin R

d times P2(Rd) where P2(R

d) denotes thespace of the probability measures over Rd with finite second moment Of coursefor an SDE with coefficients depending on (xμ) the solution Xtx

s 0 le s le t leT x isin R

d does obviously not define a flow But we see easily that if we replacethe deterministic initial condition X

txt = x isin R

d by a square integrable randomvariable X

tξt = ξ isin L2(Ft Rd)(= L2(Ft P Rd)) and consider the SDE

dXtξs = σ

(Xtξ

s PX

tξs

)dBs + b

(Xtξ

s PX

tξs

)ds

(13)s isin [t T ]Xtξ

t = ξ isin Rd

(where obviously in general Xtξ = Xtx |x=ξ ) then we have the flow property

For all 0 le t le s le T and ξ isin L2(Ft Rd) it holds Xsηr = X

tξr r isin [s T ] for

η = Xtξs This flow property should give rise to a PDE with a solution V (t ξ) =

E[(XtξT P

XtξT

)] but the fact that ξ has to belong to L2(Ft Rd) has the conse-

quence that V (t ξ) has to be considered over the Hilbert space L2(FRd) which

826 BUCKDAHN LI PENG AND RAINER

because of the absence of second-order Freacutechet differentiability even in simplestexamples makes such PDE difficult to handle As an alternative we associate withthe above SDE for Xtξ the equation

dXtxξs = σ

(Xtxξ

s PX

tξs

)dBs + b

(Xtxξ

s PX

tξs

)ds

(14)s isin [t T ]Xtxξ

t = x isin Rd

It turns out (see Lemma 31) that XtxPξs = X

txξs s isin [t T ] depends on ξ isin

L2(Ft Rd) only through its law Pξ Xtξs = X

txPξs |x=ξ and (X

txPξs X

tξs )0 le

t le s le T ξ isin L2(Ft Rd) has the flow propertyThe objective of our manuscript is to study under appropriate regularity as-

sumptions on the coefficients the second-order PDE which is associated with thisstochastic flow that is the PDE whose unique classical solution is given by thefunction

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]

(15)(t x) isin [0 T ] timesR

d ξ isin L2(Ft Rd

)

The function V is defined over [0 T ] times Rd times P2(R

d) and so the study of thefirst- and the second-order derivatives with respect to the probability law will playa crucial role In our work we have based ourselves on the notion of derivative ofa function f P2(R

d) rarr R with respect to a probability measure μ which wasstudied by Lions in his course at Collegravege de France [17] (the reader is also referredto the notes on this course redacted by Cardaliaguet [6]) The derivative of f withrespect to μ is a function partμf P2(R

d) times Rd rarr R

d (see Section 2) The mainresult of our work says that if the coefficients b and σ are twice differentiablein (xμ) with bounded Lipschitz derivatives of first and second order then thefunction V (t xPξ ) defined above is the unique classical solution of the followingnonlocal PDE of mean-field type (see Theorem 72)

parttV (t xPξ ) +dsum

i=1

partxiV (t xPξ )bi(xPξ )

+ 1

2

dsumijk=1

part2xixj

V (t xPξ )(σikσjk)(xPξ )

+ E

dsum

i=1

(partμV )i(t xPξ ξ)bi(ξPξ )(16)

+ 1

2

dsumijk=1

partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )

= 0

V (T xPξ ) = (xPξ )


with (t xPξ ) isin [0 T ]timesRd timesP2(R

d) We see in particular that in contrast to theclassical case the derivative partμV (t xPξ y) and as a second-order derivative thederivative of partμV (t xPξ y) with respect to y are involved

Mean-field SDEs also known as McKeanndashVlasov equations were discussedthe first time by Kac [13 14] in the frame of his study of the Boltzman equa-tion for the particle density in diluted monatomic gases as well as in that of thestochastic toy model for the Vlasov kinetic equation of plasma A by now classi-cal method of solving mean-field SDEs by approximation consists in the use ofso-called N -particle systems with weak interaction formed by N equations drivenby independent Brownian motions The convergence of this system to the mean-field SDE is called in the literature propagation of chaos for the McKeanndashVlasovequation

The pioneering works by Kac and in the aftermath by other authors have at-tracted a lot of researchers interested in the study of the chaos propagation andthe limit equations in different frameworks for getting an impression we refer thereader for instance to [1 3 10ndash12 15 18ndash20] and [21] as well as the referencestherein With the pioneering works on mean-field stochastic differential games byLasry and Lions (we refer to [16] and the papers cited therein but also to [6]) newimpulses and new applications for mean-field problems were given So recentlyBuckdahn Djehiche Li and Peng [4] studied a special mean field problem by apurely stochastic approach and deduced a new kind of backward SDE (BSDE)which they called mean-field BSDE they showed that the BSDE can be obtainedby an approximation involving N -particle systems with weak interaction Theycompleted these studies of the approximation with associating a kind of centrallimit theorem for the approximating systems and obtained as limit some forward-backward SDE of mean-field type which is not only governed by a Brownianmotion but also by an independent Gaussian field In [5] deepening the investi-gation of mean-field SDEs and associated mean-field BSDEs Buckdahn Li andPeng generalized their previous results on mean-field BSDEs and in a ldquoMarko-vianrdquo framework in which the initial data (t x) were frozen in the law variable ofthe coefficients they investigated the associated nonlocal PDE However our ob-jective has been to overcome this partial freezing of initial data in the mean-fieldSDE and to study the associated PDE and this is done in our present manuscriptOur approach is highly inspired by the courses given by Lions [17] at Collegravege deFrance (redacted by Cardaliaguet [6]) and by recent works of Bensoussan FrehseYam [2] and Carmona and Delarue who directly inspired by the works of LasryLions [16] and the courses of Lions [17] translated his rather analytical approachinto a stochastic one let us cite [8 9] and the references indicated therein

Let us describe the organization of our manuscript and link this description withthe explanation of the novelty in our approach

In Section 2 ldquoPreliminariesrdquo we introduce the framework of our study Partic-ular attention is paid to a recall of Lionsrsquo definition of the derivative of a function


defined over the space of probability measures over Rd with finite second mo-

ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)

In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel

A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above

2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R

d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment

intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space


P2(R2d) is endowed with the 2-Wasserstein metric

W2(μ ν) = inf(int

RdtimesRd|x minus y|2ρ(dx dy)

)12

ρ isin P2(R

2d)(21)

with ρ(middot timesR

d) = μρ(R

d times middot) = ν

μ ν isin P2

(R

d)

Among the different notions of differentiability of a function f defined overP2(R

d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R

d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property

Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)

12L2 Recall that due to the definition made

by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-

ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some

ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that

f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2

)(22)

with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R

d

such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write

f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)

] + o(|ϑ minus ϑ0|L2

)(23)

ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R

d the derivative of f P2(Rd) rarr R at

Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R

d) rarrR whichare differentiable in all elements of P2(R

d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant


K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR

d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K

∣∣y minus yprime∣∣ for all y yprime isin Rd

This motivates us to make the following definition

DEFINITION 21 We say that f isin C11b (P2(R

d)) [continuously differentiableover P2(R

d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R

d) timesRd rarr R

d is bounded and Lipschitz continuous that isthere is some real constant C such that

(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R

d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R

d) xxprime isin R

d

we consider this function partμf as the derivative of f

REMARK 21 Let us point out that if f isin C11b (P2(R

d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R

d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R

d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)

is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is

uniquely defined for all y isin Rd

Given now f isin C11b (P2(R

d)) for fixed y isin Rd the question of the differen-

tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and

it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R

d) rarr R belongs to C11b (P2(R

d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R

d)timesRd rarrR

d is a Lipschitz-continuous function forevery y isin R

d Then

part2μf (μx y) = (

partμ

((partμf )j (middot y)

)(μ x)

)1lejled

(24)(μ x y) isin P2

(R

d) timesR

d timesRd

defines a function part2μf P2(R

d) timesRd timesR

d rarrRd otimesR

d


d)) if f isin C11b (P2(R

d)) and


(i) (partμf )j (middot y) isin C11b (P2(R

d)) for all y isin Rd1 le j le d and part2

μf P2(R

d) timesRd timesR

d rarr Rd otimesR

d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R

d is differentiable for every μ isin P2(Rd) and its

derivative partypartμf P2(Rd)timesR

d rarr Rd otimesR

d is bounded and Lipschitz-continuous

Adopting the above introduced notation we consider a functionf isin C

21b (P2(R

d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ

over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int

(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes

With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled

b = (bj )1lejled isin Rd we can state now the following result

LEMMA 21 Let f isin C21b (P2(R

d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion

f (Pϑ) minus f (Pϑ0)

= E[partμf (Pϑ0 ϑ0) middot η] + 1

2E[E

[tr

(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η

)]](25)

+ 12E

[tr

(partypartμf (Pϑ0 ϑ0) middot η otimes η

)] + R(PϑPϑ0)

ϑ isin L2(FRd

)

where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)

∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)

le CE[|ϑ minus ϑ0|3]

The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf

We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)


EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin

Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr

0 ( rarr infin)

If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable

However for our purposes the above expansion is fine

PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C

21b (P2(R

d)) we have

f (Pϑ) minus f (Pϑ0) =int 1

0

d

dλf (Pϑ0+λη) dλ

(27)

=int 1

0E

[partμf (Pϑ0+ληϑ0 + λη) middot η]

dλ

Let us now compute ddλ

partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R

d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and

d

dλpartμf (Pϑ0+λη y) = d

dλpartμf (ϑ0 + λη y)

(28)= E

[part2μf (Pϑ0+ληϑ0 + λη y) middot η]

λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined

over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0

d

dλpartμf (Pϑ0+λη y) = E

[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]

(29)λ isin R y isin R

d

Consequently

d

dλpartμf (Pϑ0+ληϑ0 + λη) = E

[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]

(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η

Thus


=int 1

0E


dλ


= E[partμf (Pϑ0 ϑ0) middot η]

+int 1

0

int λ

0E

[d

dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η

]dρ dλ(211)


+int 1

0

int λ

0E

[tr

(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η

)]dρ dλ

+int 1

0

int λ

0E

[E

[tr

(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η

)]]dρ dλ

From this latter relation we get



2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr

((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)

(213)minus part2

μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η

)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr

((partypartμf (Pϑ0+ρηϑ0 + ρη)

(214)minus partypartμf (Pϑ0 ϑ0)

) middot η otimes η)]

dρ dλ

Finally from the boundedness and the Lipschitz continuity of the functions part2μf

and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf

|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement

Let us finish our preliminary discussion with two illustrating examples

EXAMPLE 22 Given two twice continuously differentiable functions h R

d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and

f (ϑ0 + η) minus f (ϑ0) =int 1

0gprime(E[

h(ϑ0 + sη)])

E[hprime(ϑ0 + sη)η

]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)


Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is

partμf (Pϑ0 y) = gprime(E[h(ϑ0)

])(partyh)(y) y isin R

d

With the same argument we see that

part2μf (Pϑ0 x y) = gprimeprime(E[

h(ϑ0)])

(partxh)(x) otimes (partyh)(y)

and

partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)

])(part2yh

)(y)

Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form

g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])

E[partyh(ϑ0) middot (ϑ minus ϑ0)

]+ 1

2gprimeprime(E[h(ϑ0)

])(E

[partyh(ϑ0) middot (ϑ minus ϑ0)

])2

+ 12gprime(E[

h(ϑ0)])

E[tr

(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)

)]+ O

((E

[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])

EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have

D2ϑ f (ϑ0)(η η) = Dϑ

[Dϑf (ϑ)

](η)|ϑ=ϑ0 = minusE

[eiϑ0

]η2

Then

R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1

2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2

as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable


3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that

(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R

k) = Pϑϑ isin L2(F0Rk) k ge 1

By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0

Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R

dtimesd and b R

d times P2(Rd) rarr R

d we consider for the initial data (t x) isin [0 T ] times Rd and

ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)

Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]

We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b

E[

supsisin[tT ]

∣∣Xtxξs minus Xtxprimeξ

s

∣∣2]le C

∣∣x minus xprime∣∣2(33)

for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-

ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude

Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)

Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(

XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)


for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X

tξs isin

L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)

and (s η) respectively

Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]

we get from the uniqueness of the solution that Xsηr = X

tξr r isin [s T ] and con-

sequently XsX

txξs η

r = Xtxξr r isin [t T ] that is we have (35)

Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R

d) rarrR an associated value function

V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)

and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR

d timesP2(Rd) rarrR For

this the following lemma is useful

LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate

E[

supsisin[tT ]

∣∣Xtx1ξ1s minus Xtx2ξ2

s

∣∣p]le Cp

(|x1 minus x2|p + W2(Pξ1Pξ2)p)

(37)

for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)

PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have

W2(PϑPθ ) = inf(

E[∣∣ϑ prime minus θ prime∣∣2])12

for all ϑ prime θ prime isin L2(F0Rd)

with Pϑ prime = PϑPθ prime = Pθ

(38)

le (E

[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)

because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R

d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b

E[

supsisin[tT ]

∣∣Xtξ1s minus Xtξ2

s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(

Ft Rd) t isin [0 T ]


and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients

E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)

for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from

the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2

supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]

∣∣Xtξ prime1ξ1

s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E

[∣∣ξ prime1 minus ξ prime

2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]

for all ξ prime1 ξ

prime2 isin L2(Ft Rd) with Pξ prime

1= Pξ1 and Pξ prime

2= Pξ2 Hence taking at the

right-hand side of (311) the infimum over all such ξ prime1 ξ

prime2 isin L2(Ft Rd) and con-

sidering the above characterization of the 2-Wasserstein metric we get

supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)

v isin [t T ] Then Gronwallrsquos inequality implies

supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s

)2 le CW2(Pξ1Pξ2)2

(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)

which allows to deduce from the estimate (310)

E[

supsisin[tT ]


s

∣∣p]le Cp


(314)

for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now

REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR

d the processes Xtxξ1 and Xtxξ2 are indistinguishable


whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define

XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(

Ft Rd)(315)

and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X

txξs =

XtxPξs = X

txξs s isin [t T ] (t x) isin [0 T ] times R

d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later

4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R

d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X

txξs isin L2(FsRd) s isin [t T ] For this we suppose

the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C

11b (Rd times

P2(Rd) rarr R

dtimesd times Rd) that is the components σij bj 1 le i j le d have the

following properties

(i) σij (x middot) bj (x middot) belong to C11b (P2(R

d)) for all x isin Rd

(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R

d)(iii) The derivatives partxσij partxbj Rd timesP2(R

d) rarr Rd and partμσij partμbj Rd times

P2(Rd) timesR

d rarr Rd are bounded and Lipschitz continuous

Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x

LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above

Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with

respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX

txPξ isin S2([t T ]Rdtimesd) such that

E[

supsisin[tT ]

∣∣Xtx+hPξs minus X

txPξs minus partxX

txPξs h

∣∣2]= o

(|h|2) as Rd h rarr 0

[Recall that o(|h|) stands for an expression which tends quicker to zero than

h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi

XtxPξ

sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE

partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]


1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX

txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R

d and ξ ξ prime isin L2(Ft Rd)

(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup

sisin[tT ]∣∣partxX

txPξs minus partxX

txprimePξ primes

∣∣p]le Cp

(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)

PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX

tξs

)

and b(s x) = b(xPX

tξs

) and taking into account that these coefficients are Lip-

schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ

and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ

REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX

tξPξs = partxX

txPξs |x=ξ s isin [t T ] is the unique solu-

tion in S2([t T ]Rdtimesd) of the SDE

partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX

tξPξr dr s isin [t T ]

Moreover it satisfies

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs

∣∣p]le Cp P -as(44)

for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb

Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension

Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative

Y txξs (η) = L2 minus lim

hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)


exists we consider the SDE

Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr

s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form

Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr

s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X

tξ+hηs is given by

parthXtξ+hηs |h=0 = partxX

txPξs |x=ξ middot η + Y txξ

s (η)|x=ξ (48)

Recalling that for some real C independent of (t xPξ )

E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C

we see that for partxXtξPξs = partxX

txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation

Y tξs (η) = Y txξ

s (η)|x=ξ (410)

we get by formal differentiation of the above equation

Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dr s isin [t T ]

As concerns the above term (Dϑσ )(Xtxξr X

tξr )(partxX

tξPξs η + Y

tξs (η)) we see

from the definition of the differentiability of σ with respect to the probability mea-


sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR

Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-

nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of

the SDE for XtxPξ but governed by B instead of B

Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]

Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent

copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-

ready introduced partxXtxPξr is the L2(P )-derivative of X

txPξr with respect to x

partxXtξ Pξr = partxX

txPξr |x=ξ and

Y txξs (η) = L2(P ) minus lim

hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)

Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that

(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX

tξ Pξs η + Y t ξ

s (η))]

(414)

s isin [t T ]Using also the corresponding formula for (Dϑb)(X

txξs X

tξr )(partxX

tξPξs η +

Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =

partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain

Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX

tξ Pξr η + Y t ξ

r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE

Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr

s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that

E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]

η isin L2(Ft )(417)

for some constant C independent of (t xPξ ) This shows in particular that

Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)

Y txξs (middot) isin L

(L2(Ft )L

2(Fs)) s isin [t T ]

We are now able to show in a rigorous manner that our process Ytxξs (η) is the

directional derivative of Xtxξs in direction η

LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)

that is Ytxξs (η) is the directional derivative of X

txξs in direction η isin L2(Ft )

PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that

σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ

s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)

we have for s isin [t T ]partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (

Xtξ+hηs minus Xtξ

s

)]

Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ

E[∣∣βs(xh) minus (partμσ)

(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs

)2 + E[∣∣Xtξ+hη

s minus Xtξs

∣∣2])dλ

le CE[∣∣Xtξ+hη

s minus Xtξs

∣∣2](421)

le CE[E

[∣∣XtyPξ+hηs minus X

typrimePξs

∣∣2]|y=ξ+hηyprime=ξ

]le CE

[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ

]le Ch2E

[η2]

s isin [t T ] x h isinR ξ η isin L2(Ft )

Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that

E[∣∣αs(xh) minus (partxσ )

(X

txPξs P

Xtξs

)∣∣p]le Cp

(E

[∣∣XtxPξ+hηs minus X

txPξs

∣∣p] + W2(PXtξ+hηs

PX

tξs

)p)

(422)le CpW2(Pξ+hηPξ )

p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2

s isin [t T ] x h isin R ξ η isin L2(Ft )


Recall that with the processes α(xh) and β(xh) the difference between

XtxPξ+hηs and X

txPξs writes for all s isin [t T ] as follows

XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X

txPξ+hηr minus XtxPξ

)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr

Consequently using the equation for Y txξ (η) we have

XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X

txPξr minus hY txξ

r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη

r minus Xtξr minus h

(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t

(αr(xh) minus (partxσ )

(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE

[(βr(xh) minus (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr

From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain

E[∣∣αr(xh) minus (partxσ )

(X

txPξr P

Xtξr

)∣∣2∣∣XtxPξ+hηr minus X

txPξr

∣∣2](425)

le Ch2E[η2]

W2(Pξ+hηPξ )2 le Ch4E

[η2]2

and (421) yields

E[∣∣E[(

βr(h x) minus (partμσ)(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E

[∣∣βr(h x) minus (partμσ)(X

txPξr P

Xtξr

Xt ξr

)∣∣2](426)

times E[∣∣Xtξ+hη

r minus Xtξr

∣∣2]]le Ch4E

[η2]2

r isin [t T ] h x isin R ξ η isin L2(Ft )

Consequently the remainder R1(s xh) can be estimated as follows

E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2

h x isin R ξ η isin L2(Ft )


and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]

supxisinR

E[

supsisin[tu]

∣∣XtxPξ+hηs minus X

txPξs minus hY txξ

s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E

[∣∣Xtξ+hηr minus Xtξ

r minus h(partxX


r (η))∣∣]2]

dr

In order to conclude let us now estimate

E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]

For this we observe that

Xtξ+hηr minus Xtξ

r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)

where I1(r h) = (XtxPξ+hηr minus X

txPξr minus hY

txξr (η))|x=ξ satisfies

E[∣∣I1(r h)

∣∣]2 le supxisinR

E[∣∣XtxPξ+hη

r minus XtxPξr minus hY txξ

r (η)∣∣2]

and for I2(r h) = Xtξ+hηPξ+hηr minus X

tξPξ+hηr minus hpartxX

tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E

[∣∣partxXtξ+λhηPξ+hηr minus partxX

tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E

[∣∣partxXtyPξ+hηr minus partxX

typrimePξr

∣∣2]|y=ξ+λhηyprime=ξ

]dλ

le Ch4E[η2]2

Therefore combining these three latter estimates we obtain

E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2

for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R

and ξ isin L2(Ft ) The proof is complete now

PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)

) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)


be the unique solution of the system of (uncoupled) SDEs

Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partxXtyPξr

(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot U t ξr (y)

]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr

s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative

Ytxξs (η) of ξ rarr X

txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ

s (ξ ) middot η] s isin [t T ] P -as(431)

REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)

(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX

tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate

Before proving the above proposition let us state the following lemma


LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R

such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)

(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]

∣∣Utxξs (y) minus Utxprimeξ prime

s

(yprime)∣∣p]

(432)

le Cp

(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)

REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)

depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)

Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that

Utξs (y) = U

txPξs (y)|x=ξ s isin [t T ] P -as(433)

The same argument allows also to substitute Ft -measurable random variables fory in U

tξs (y)

PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX

txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )

E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)

As concern the proof of the estimate

E[

supsisin[tT ]

(∣∣Utxξs (y) minus Utxprimeξ prime

s

(yprime)∣∣p + ∣∣Utξ

s (y) minus Utξ primes

(yprime)∣∣p)]

(435) le Cp


we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ

E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer

) middot partxXtyprimePξ primer

]∣∣p](436)

le Cp

(E

[∣∣XtxPξr minus X

txprimePξ primer

∣∣2p + (E

[∣∣XtyPξr minus X

typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E

[∣∣partxXtyPξs minus partxX

typrimePξ primes

∣∣2])p2


Once having the above estimate we can use our estimates for XtxPξ andpartxX

txPξ in order to deduce that

E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr


minus (partμσ)(Xtξ prime

r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp


Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates

and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp

such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )

E[

suprisin[ts]

∣∣Utxξr (y) minus Utxprimeξ prime

r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t

∣∣E[∣∣U t ξr (y) minus U t ξ prime

r

(yprime)∣∣2]∣∣p2

dr

] s isin [t T ]

Then from this estimate for p = 2

E[

suprisin[ts]

∣∣Utξr (y) minus Utξ prime

r

(yprime)∣∣2]

= E[E

[sup

risin[ts]∣∣Utxξ

r (y) minus Utxprimeξ primer

(yprime)∣∣2]

|x=ξxprime=ξ prime]

(440)le C

(E

[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E

[int s

tE

[∣∣U t ξr (y) minus U t ξ prime

r

(yprime)∣∣2]

dr

]

s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof

We are now able to give the proof of Proposition 41

PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )


which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)

In order to obtain the wished relation we substitute in the equation for Utξ (y)

for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields

E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ

r (ξ ) middot η]dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ

r (ξ ) middot η]dr

+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr

) middot partxXtξ Pξr middot η]]

dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ

r (ξ ) middot η]]dBr

+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ

r (ξ ) middot η]]dr s isin [t T ]

Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that

E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot partxXtξ Pξr middot η]

and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form

E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η + E

[U t ξ

r (ξ ) middot η])]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ

r (ξ ) middot η])]dr s isin [t T ]


But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that

Y tξs (η) = E

[Utξ

s (ξ ) middot η] s isin [t T ](442)

Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields

E[U

txPξs (ξ ) middot η]=

int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U

txPξr (ξ ) middot η]

dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX

tξ Pξr middot η + Y t ξ

r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr

s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that

YtxPξs (η) = E

[U

txPξs (ξ ) middot η]

s isin [t T ] η isin L2(Ft )(444)

The proof is complete now

The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case

PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping

L2(Ft ) ξ rarr Xtxξs = X

txPξs isin L2(Fs)

is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies

DξXtxξs (η) = Y txξ

s (η) = E[U



REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-

tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of

XtxPξs with respect to the probability law Pξ by putting

partμXtxPξs (y) = U

txPξs (y) s isin [t T ] x y isinR


We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX

txξs (η) = Y txξ

s (η) = E[partμX



PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y

txξs (η) of L2(Ft ) ξ rarr

Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y

txξs (middot) isin

L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )

L2(Fs)) Then using the preceding lemma since

E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U

txPξs (ξ ) middot η]∣∣2]

le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )

for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs

∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)

∣∣2] η isin L2(Ft ) with E[η2] le 1

le C

Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]

E[∣∣Y txξ

s (η) minus Y txξ primes (η)

∣∣2] = E[∣∣E[(

UtxPξs (ξ ) minus U

txPξ primes (ξ )

) middot η]∣∣2]le E

[E

[∣∣UtxPξs (ξ ) minus U

txPξ primes (ξ )

∣∣2]E

[η2]]

le C2W2(Pξ Pξ prime)2E[η2]

η isin L2(Ft )

that is∥∥Y txξs minus Y txξ prime

s

∥∥2L(L2L2)

= supE

[∣∣Y txξs (η) minus Y txξ prime

s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1

le CW2(Pξ Pξ prime)2 le CE

[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )

This proves that the directional derivative Ytxξs is a bounded operator (and hence

a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X

txξs with respect to ξ The proof is complete

A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult

THEOREM 41 Let (σ b) isin C11b (Rd times P2(R

d) rarr Rdtimesd times R

d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R

d the mapping


L2(Ft Rd) ξ rarr Xtxξs = X

txPξs isin L2(FsRd) is Freacutechet differentiable with

Freacutechet derivative

DξXtxξs (η) = E

[U

txPξs (ξ ) middot η] =

(E

[dsum

j=1

UtxPξ

sij (ξ ) middot ηj

])1leiled

(447)

for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =

((UtxPξ

sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE

UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr

) middot Utξrkj (y)

]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr

s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2

F(t T

Rdtimesd) is that of the SDE

Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr

) middot Utξrkj (y) dB

r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr

) middot Utξrkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr

s isin [t T ]1 le i j le d


REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R

d) rarr R explained in Section 2 we con-

sider UtxPξs (y) = (U

txPξ

sij (y))1leijled as derivative over P2(Rd) of X

txPξs =

(XtxPξ

si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]

partμXtxPξs (y) = (

partμXtxPξ

sij (y))1leijled = U

txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled

5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper

Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R

d) rarrRdtimesd timesR

d) that

is (σ b) isin C11b (Rd times P2(R


d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties

(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d

(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR

d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz

REMARK 51 With the existence of the second-order mixed derivativespartxl

(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R

d timesP2(Rd) timesR

d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement

LEMMA 51 Let g isin C21b (Rd timesP2(R


d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d

partxl

(partμg(xμy)

) = partμ

(partxl

g(xμ))(y) (xμ y) isin R

d timesP2(R

d) timesRd

PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)

I = (g(x + zPξ+η) minus g(x + zPξ )

) minus (g(xPξ+η) minus g(xPξ )

)=

int 1

0E

[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)

)η]ds

=int 1

0

int 1

0E

[partx(partμg)(x + tzPξ+sη ξ + sη)ηz

]ds dt

= E[partx(partμg)(xPξ ξ)ηz

] + R1((x ξ) (z η)

)


with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time

I = (g(x + zPξ+η) minus g(xPξ+η)

) minus (g(x + zPξ ) minus g(xPξ )

)=

int 1

0

((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )

)dt middot z

=int 1

0

int 1

0E

[partμ

((partxg)(x + tzPξ+sη)

)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)

with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-

chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence

partx(partμg)(xPξ y) = partμ

((partxg)(xPξ )

)(y) Pξ (dy)-ae

Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have

partx(partμg)(xPξ+εθ y) = partμ

((partxg)(xPξ+εθ )

)(y) Pξ+εθ (dy)-ae

and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields



)(y) for all y isin R

Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0


((partxg)(xPξ )

)(y) for all x y isinR ξ isin L2(F)

The proof is complete

After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations

We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique

solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX

txPξr dBr s isin [t T ](51)

(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2

xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of


the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr

)2)dBr s isin [t T ]

We also notice that with standard arguments we obtain the following lemma

LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )

(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup

sisin[tT ]∣∣part2

xXtxPξs minus part2

xXtxprimePξ primes

∣∣p]le Cp


In the same manner as one proves the L2-differentiability of x rarr XtxPξs

s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX

txPξs (y) s isin [t T ]

Standard arguments give the following result

LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE

partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX

txPξr (y) middot partxX

txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr


respectively where the latter equation is coupled with the SDE

partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]

which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR

of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for

all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )

(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

)∣∣p + ∣∣party

(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx

(partμXtxprimePξ prime (yprime))∣∣p

(57)+ ∣∣party

(partμXtxPξ (y)

) minus party

(partμXtxprimePξ prime (yprime))∣∣p)]

le Cp


Let us now consider the derivative of the process partxXtxPξ with respect to the

probability law Knowing already that the mixed second-order derivatives partxpartμ

and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing

LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )

partμpartxXtxPξs (y) = partxpartμX


PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ

are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ

shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0

E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs

) minus partxXtxPξs

∣∣∣∣2]le Ch2(58)


Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0

E

[∣∣∣∣1

λ

(partxX

txPξ+ληs minus partxX

txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2]

le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X

tx+hPξ+ληs minus X

tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX

tx+hPξ+vηs (ξ + vη)

minus partμXtxPξ+vηs (ξ + vη)

) middot η]dv minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX

tx+uPξ+vηs (ξ + vη)

) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η∣∣]dudv

∣∣∣∣2]

Thus taking into account that

E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)

le C(|u| + W2(Pξ+vηPξ )

)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0

But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of

partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in

the preceding section that this directional derivative is in fact a Freacutechet derivative

Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]


of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX

txPξs s isin [t T ] The state-

ment of the lemma follows directly

It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE

Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partxXtyPξr(511)

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr

Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted

process Utxξs (y) = U

txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction

η isin L2(Ft ) This gives for the formal directional derivative

Ztxξs (y η) = L2 minus lim

hrarr0

1

h

(Utxξ+hη

s (y) minus Utxξs (y)

) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot E[partμpartxX

tyPξr (ξ ) middot η]]

dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ

r (ξ ) middot η])]]dBr(513)

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ

r (ξ ) middot η])]]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot U t ξr (y) middot (

partxXtξ Pξr middot η

+ E[U t ξ


s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point

out that in the above equation we have used the notation (Xtξ

Utξ

(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the

process Xtξ

is the solution of the SDE for Xtξ and Utξ

that of the SDE for Utξ but both with the data (ξB) instead of (ξB)

Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)

in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is

defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for

ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)

Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ

)(xPϑ yϑ) middot θ]


Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X

tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ

r (ξ ) middot η])]+ party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot E[U

tyPξr (ξ ) middot η]

η isin L2(Ft )

r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section

Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2

F(t T ) defined as the unique solution of the following SDE

UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y) middot UtxPξ

r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


] middot UtxPξr (z) dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX

tyPξr middot partxX

tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r

) middot partxXtyPξr U

tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot U t ξr (y z)

]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ

r (y) middot partxXtzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r

) middot U t ξr (y) middot Utξ

r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot U t ξr (y) middot partxX

tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot U t ξr (y) middot U t ξ

r (z)]dBr

s isin [0 T ]combined with the SDE for Utξ (y z) = (U

tξs (y z))sisin[tT ] obtained by sub-

stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U

tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking

the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)

and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in

the preceding section) and from the uniqueness of the solution of these SDEs weconclude that

Ztxξs (y η) = E

[U

txPξs (y ξ ) middot η]

s isin [t T ] y isin R(516)

We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates

LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )

(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]

∣∣UtxPξs (y z) minus U

txprimePξ primes

(yprime zprime)∣∣p]

(517)

le Cp

(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)


The estimates of Lemma 55 allow to show in analogy to the argument devel-

oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X

txPξs that

the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U

tξs (y) isin

L2(Fs) are Freacutechet differentiable and

Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)

= partxUtxPξs (y)|x=ξ middot η + E

[Utξ

s (y ξ ) middot η]

But this means that

part2μX

txPξs (y z) = U

txPξs (y z) s isin [0 T ](518)

for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives

and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations

THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX

txPξs

1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and

y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi

XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2

μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such

that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and

Dξ [partμXtxPξs (y)](middot) satisfy

(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)

s isin [t T ] y isin Rd

Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi

XtxPξ )(y) coin-cide that is

partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs

)(y) s isin [t T ] P -as

and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2

μXtxPξs (y z) partyi

(partμX

txPξs (y)

))


1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)

(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]

∣∣MtxPξs (y z) minus M

txprimePξ primes


(520)

le Cp


6 Regularity of the value function Given a function isin C21b (Rd times

P2(Rd)) the objective of this section is to study the regularity of the function

V [0 T ] timesRd timesP2(R

d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]

(61)(t x ξ) isin [0 T ] timesR

d times L2(Ft Rd

)

LEMMA 61 Suppose that isin C11b (Rd timesP2(R

d)) Then under our Hypoth-

esis (H1) V (t middot middot) isin C11b (Rd times P2(R

d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi

V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x

Pξ y))1leiled are of the form

partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)

(partμV )i(t xPξ y) =dsum

j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]

Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R


(i)∣∣partxi

V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)

∣∣ le C

(ii)∣∣partxi

V (t xPξ ) minus partxiV

(t xprimePξ prime

)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i

(t xprimePξ prime yprime)∣∣

(64)le C

(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))

(iii)∣∣V (t xPξ ) minus V

(t prime xPξ

)∣∣ + ∣∣partxiV (t xPξ ) minus partxi

V(t prime xPξ


(t prime xPξ y

)∣∣ le C∣∣t minus t prime

∣∣12


PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use

In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the

same sense Xtxξ = XtxPξ Then (XtxξT X

tξT ) = (X

txPξ

T PX

tξT

)

As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz

continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X

txξT X

tξT )) of the

mapping L2(Ft ) ξ rarr (XtxξT X

tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have

Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ

T (ξ ) middot η](65)

+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ

T (ξ ) middot η]+ E

[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ

T middot η + E[partμX

tξT (ξ ) middot η])]

(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X

txξT P

XtξT

) with respect to the measure in Pξ

is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxXtyPξ

T

+ partμ(X

txPξ

T PX

tξT

Xt ξT

) middot partμXtξ Pξ

T (y)] y isin R

In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp



As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X

txPξ

T PX

tξT

)] is Freacutechet differentiable and for all η isin L2(Ft )

Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is

partμV (t xPξ y) = E[partμ

(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)

But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)

As concerns the derivative of V (t xPξ ) = E[(XtxPξ

T PX

tξT

)] with respect

to x since z rarr (zPX

tξT

) is a (deterministic) function with a bounded Lipschitz

continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections

In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt

s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime

solve thefollowing SDEs (for simplicity we put b = 0 again)

Xtξ primes+t = ξ prime +

int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)

XtxPξ primes+t = x +

int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt

r s isin [0 T minus t](611)

Consequently (XtxPξ primemiddot+t X

tξ primemiddot+t ) and (X0xPξ prime X0ξ prime

) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-

tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X

tξ primemiddot+t ) and

(X0xPξ prime X0ξ prime) coincide and hence

V (t xPξ ) = V (t xPξ prime) = E[

(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X

0xPξ primeT minust P

X0ξ primeT minust

)]


Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V

(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X

0xPξ primeT minust prime P

X0ξ primeT minust prime

)∣∣](613)

le C(E

[∣∣X0xPξ primeT minust minus X

0xPξ primeT minust prime

∣∣] + W2(PX

0ξ primeT minust

PX

0ξ primeT minust prime

))

le C(E



∣∣2 + ∣∣X0ξ primeT minust minus X


∣∣2])12

But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime

we get∣∣V (t xPξ ) minus V(t prime xPξ


∣∣12(614)

The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX

txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime

(y) = (Xtξ prime partxX

tξ primeUtξ prime

(y)) y isin R [see (32) (51) (429)] we seeagain that ((

NtxPξ primemiddot+t (y)

)xyisinR

(N

tξ primemiddot+t (y)yisinR

))and ((

N0xPξ prime (y))xyisinR

(N0ξ prime

(y)yisinR))

are equal in lawHence from (69) we deduce


(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]

Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX

0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV

(t prime xPξ y

)∣∣le C

(E



∣∣ + ∣∣X0yPξ primeT minust minus X

0yPξ primeT minust prime

∣∣ + ∣∣X0ξ primeT minust minus X


∣∣(616)

+ ∣∣partxX0yPξ primeT minust minus partxX


∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX

0xPξ primeT minust prime (y)

∣∣+ ∣∣partμX

0ξ primePξ primeT minust (y) minus partμX

0ξ primePξ primeT minust prime (y)

∣∣] + W2(PX

0ξ primeT minust

PX


))


Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV

(t prime xPξ y

)∣∣le C middot sup

xyisinR(E



∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX


∣∣2(617)

+ ∣∣partμX0xPξ primeT minust (y) minus partμX


∣∣2])12

and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX

0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV

(t prime xPξ y


∣∣12(618)

The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here

Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )

LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C

21b (Rd times P2(R

d)) Then for all t isin [0 T ]V (t middot middot) isin C

21b (Rd times P2(R

d)) and the mixed second-order derivatives are sym-metric

partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)

(t x y) isin [0 T ] timesRd timesR

d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)

part2μV (t xPξ y z) party

(partμV (t xPξ y)

))

there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd

ξ ξ prime isin L2(Ft Rd)

(i)∣∣U(t xPξ y z)

∣∣ le C

(ii)∣∣U(t xPξ y z) minus U

(t xprimePξ prime yprime zprime)∣∣

(619)le C

(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))

(iii)∣∣U(t xPξ y z) minus U

(t prime xPξ y z


∣∣12

PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation


for the second-order derivative with respect to the measure part2μV (t xPξ y z) and

to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2

xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of

and of the processes XtxPξ and Xtξ shows that

partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)

(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )

For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)

in Lemma 61 as well as (69) and (66) We observe that

partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)

with

V1(t xPξ y) = E[(partx)

(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]

Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x

Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of

ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)

(t x ξ) isin [0 T ] timesRtimes L2(Ft )

that for all η isin L2(Ft )

DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]


(recall the notation introduced in Section 4) On the other hand we know alreadythat

(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ

T (y)) middot η + E

[(part2μX

tξ Pξ

T

)(y ξ ) middot η]

Consequently we have

DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ

T (ξ ))]] middot η]

Therefore

partμV3(t xPξ y z)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT


T (y) middot partμXtxPξ

T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]


t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation

DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]

(626)

that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)

combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3

(t xprimeP prime

ξ yprime zprime)∣∣

(627)le C


for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3

(t prime xPξ y z


∣∣12(628)

for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)

we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2

μV (t xPξ y z) The proof is complete

7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE

THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin

C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin

Rd times P2(R

d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR

d times P(Rd) (for short F isin C1(21)b ([0 T ] times R

d times P2(Rd))) Then under Hy-

pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc

formula is satisfied

F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]

PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension

Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin

L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have

f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)

)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2

μf and partypartμf

∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni

)32 0 le i le 2n minus 1


Thus taking into account the relations (recall that B and B are independent Brow-nian motions)

(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

(iii) E[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni

with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr

(XtxPξr P

Xtξr

) isin L2(FRtimesP2(R)) we get from the above sum over the second-


order Taylor expansion as n rarr +infin

f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]

From this latter relation we see that for fixed (t ξ) the function s rarr f (PX

tξs

) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular

partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]

Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times

RtimesP2(R)) also (s x) = F(s xPX

tξs

) is continuously differentiable

parts(s x) = (partsF )(s xPX

tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]

and twice continuously differentiable with respect to x Hence we can apply to

(sXtxPξs )(= F(sX

txPξs P

Xtξs

)) the classical Itocirc formula This yields

F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]


The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR

d timesP2(Rd)

LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-

sis (H2) V isin C1(21)([0 T ] times Rd times P2(R

d)) and its derivative parttV (t xPξ )

with respect to t verifies for some constant C isin R

(i)∣∣parttV (t xPξ )

∣∣ le C

(ii)∣∣parttV (t xPξ ) minus parttV

(t xprimePξ prime

)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)

)(77)

(iii)∣∣parttV (t xPξ ) minus parttV

(t prime xPξ


∣∣12

for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)

PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed

without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)

Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again

V (t xPξ ) minus V (T xPξ )

=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr


Then it is evident that V (t xPξ ) is continuously differentiable with respect to t

parttV (t xPξ ) = minusE[partx

(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]

Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete

Now we are able to establish and to prove our main result

THEOREM 72 We suppose that isin C21b (Rd times P2(R

d)) Then under

Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ

T PX

tξT

)] (t x ξ) isin[0 T ]timesR

d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R

d))

of the PDE

0 = parttV (t xPξ ) +dsum

i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1

(partμV )i(t xPξ ξ)bi(ξPξ )

(711)

+ 1

2

dsumijk=1


]

(t x ξ) isin [0 T ] timesRd times L2(

Ft Rd)

V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(

FRd)

PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property

(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)

(712)0 le t le s le r le T x isin R ξ isin L2(Ft )


of our dynamics as well as

V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)

s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that

V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]

that is V (sXtxPξs P

Xtξs

) s isin [t T ] is a martingale On the other hand since

due to Lemma 71 the function V isin C1(21)b ([0 T ] times R

d times P2(Rd)) satisfies the

regularity assumptions for the Itocirc formula we know that

V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs

) minus V (t xPξ )(716)

=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]


from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in

the class C1(21)b ([0 T ]timesR

d timesP2(Rd)) Let U isin C

1(21)b ([0 T ]timesR

d timesP2(Rd))

be a solution of PDE (711) Then from the Itocirc formula we have that

U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr

s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )

U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )

This proves that the functions U and V coincide that is the solution is unique inC

1(21)b ([0 T ] timesR

d timesP2(Rd)) The proof is complete

Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited

REFERENCES

[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193

[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1

[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849

[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754

[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268

[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia

[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957

[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710

[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471

[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884


[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186

[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876

[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985

[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849

[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250

[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621

[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp

[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299

[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint

[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844

[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185

R BUCKDAHN

LABORATOIRE DE MATHEacuteMATIQUES

CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE

6 AVENUE VICTOR-LE-GORGEU

BP 809 29285 BREST CEDEX

FRANCE

AND

SCHOOL OF MATHEMATICS

SHANDONG UNIVERSITY

JINAN SHANDONG PROVINCE 250100P R CHINA

E-MAIL rainerbuckdahnuniv-brestfr

J LI

SCHOOL OF MATHEMATICS AND STATISTICS

SHANDONG UNIVERSITY WEIHAI

WEIHAI SHANDONG PROVINCE 264209P R CHINA

E-MAIL juanlisdueducn

S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE

E-MAIL catherineraineruniv-brestfr

Introduction

Preliminaries

The mean-ﬁeld stochastic differential equation

First-order derivatives of XtxPxi

Second-order derivatives of XtxPxi

Regularity of the value function

Itocirc formula and PDE associated with mean-ﬁeld SDE

Acknowledgments

References

Authors Addresses


7 Itocirc formula and PDE associated with mean-field SDE 870Acknowledgments 877References 877

1 Introduction Given a complete probability space (FP ) endowedwith a Brownian motion B = (Bt )tisin[0T ] and its filtration F = (Ft )tisin[0T ] aug-mented by all P -null sets and a sufficiently rich sub-σ -algebra F0 independentof B we consider the mean-field stochastic differential equation (SDE) alsoknown under the name of McKeanndashVlasov SDE

dXtxs = σ

(Xtx

s PXtxs

)dBs + b

(Xtx

s PXtxs

)ds

(11)s isin [t T ]Xtx

t = x isinRd

It is well known that under an appropriate Lipschitz assumption on the coeffi-cients this equation possesses for all (t x) isin [t T ]timesR

d a unique solution Xtxs s isin

[0 T ] For the classical SDE whose coefficients σ(xμ) = σ(x) b(xμ) = b(x)

depend only on x isin Rd but not on the probability measure μ it is well known

that the solution Xtxs 0 le t le s le T x isinR

d defines a flow and if the coefficientsare regular enough the unique classical solution of the partial differential equation(PDE)

parttV (t x) + 12 tr

(σσ lowast(x)D2

xV (t x)) + b(x)DxV (t x) = 0

(t x) isin [0 T ] timesRd(12)

V (T x) = (x) x isin Rd

is V (t x) = E[(XtxT )] (t x) isin [0 T ] times R

d But how about the above SDEwhose coefficients depend on (xμ) isin R

d times P2(Rd) where P2(R

d) denotes thespace of the probability measures over Rd with finite second moment Of coursefor an SDE with coefficients depending on (xμ) the solution Xtx

s 0 le s le t leT x isin R

d does obviously not define a flow But we see easily that if we replacethe deterministic initial condition X

txt = x isin R

d by a square integrable randomvariable X

tξt = ξ isin L2(Ft Rd)(= L2(Ft P Rd)) and consider the SDE

dXtξs = σ

(Xtξ

s PX

tξs

)dBs + b

(Xtξ

s PX

tξs

)ds

(13)s isin [t T ]Xtξ

t = ξ isin Rd

(where obviously in general Xtξ = Xtx |x=ξ ) then we have the flow property

For all 0 le t le s le T and ξ isin L2(Ft Rd) it holds Xsηr = X

tξr r isin [s T ] for

η = Xtξs This flow property should give rise to a PDE with a solution V (t ξ) =

E[(XtξT P

XtξT

)] but the fact that ξ has to belong to L2(Ft Rd) has the conse-

quence that V (t ξ) has to be considered over the Hilbert space L2(FRd) which



dXtxξs = σ

(Xtxξ

s PX

tξs

)dBs + b

(Xtxξ

s PX

tξs

)ds


t = x isin Rd




txPξs |x=ξ and (X

txPξs X

tξs )0 le



V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d ξ isin L2(Ft Rd

)




d) times Rd rarr R



i=1


+ 1

2

dsumijk=1

part2xixj


+ E

dsum

i=1


+ 1

2

dsumijk=1


= 0



















W2(μ ν) = inf(int


)12

ρ isin P2(R

2d)(21)


d) = μρ(R


μ ν isin P2

(R

d)










)(22)


d




)(23)














d) timesRd rarr R




d) xxprime isin R

d














d)timesRd rarrR


d Then


partμ


)(μ x)

)1lejled


(R

d) timesR

d timesRd


d) timesRd timesR

d rarrRd otimesR

d



d)) and




μf P2(R

d) timesRd timesR

d rarr Rd otimesR




d rarr Rd otimesR



21b (P2(R










2E[E

[tr


)]](25)

+ 12E

[tr


)] + R(PϑPϑ0)

ϑ isin L2(FRd

)


∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)







0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses



dXtxξs = σ

(Xtxξ

s PX

tξs

)dBs + b

(Xtxξ

s PX

tξs

)ds


t = x isin Rd




txPξs |x=ξ and (X

txPξs X

tξs )0 le



V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d ξ isin L2(Ft Rd

)




d) times Rd rarr R



i=1


+ 1

2

dsumijk=1

part2xixj


+ E

dsum

i=1


+ 1

2

dsumijk=1


= 0



















W2(μ ν) = inf(int


)12

ρ isin P2(R

2d)(21)


d) = μρ(R


μ ν isin P2

(R

d)










)(22)


d




)(23)














d) timesRd rarr R




d) xxprime isin R

d














d)timesRd rarrR


d Then


partμ


)(μ x)

)1lejled


(R

d) timesR

d timesRd


d) timesRd timesR

d rarrRd otimesR

d



d)) and




μf P2(R

d) timesRd timesR

d rarr Rd otimesR




d rarr Rd otimesR



21b (P2(R










2E[E

[tr


)]](25)

+ 12E

[tr


)] + R(PϑPϑ0)

ϑ isin L2(FRd

)


∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)







0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses


















W2(μ ν) = inf(int


)12

ρ isin P2(R

2d)(21)


d) = μρ(R


μ ν isin P2

(R

d)










)(22)


d




)(23)














d) timesRd rarr R




d) xxprime isin R

d














d)timesRd rarrR


d Then


partμ


)(μ x)

)1lejled


(R

d) timesR

d timesRd


d) timesRd timesR

d rarrRd otimesR

d



d)) and




μf P2(R

d) timesRd timesR

d rarr Rd otimesR




d rarr Rd otimesR



21b (P2(R










2E[E

[tr


)]](25)

+ 12E

[tr


)] + R(PϑPϑ0)

ϑ isin L2(FRd

)


∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)







0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses









d) timesRd rarr R




d) xxprime isin R

d














d)timesRd rarrR


d Then


partμ


)(μ x)

)1lejled


(R

d) timesR

d timesRd


d) timesRd timesR

d rarrRd otimesR

d



d)) and




μf P2(R

d) timesRd timesR

d rarr Rd otimesR




d rarr Rd otimesR



21b (P2(R










2E[E

[tr


)]](25)

+ 12E

[tr


)] + R(PϑPϑ0)

ϑ isin L2(FRd

)


∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)







0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses




μf P2(R

d) timesRd timesR

d rarr Rd otimesR




d rarr Rd otimesR



21b (P2(R










2E[E

[tr


)]](25)

+ 12E

[tr


)] + R(PϑPϑ0)

ϑ isin L2(FRd

)


∣∣ le C((

E[|ϑ minus ϑ0|2])32 + E

[|ϑ minus ϑ0|3])(26)







0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses




0 ( rarr infin)




21b (P2(R

d)) we have


0

d


(27)

=int 1

0E


dλ




d



(28)= E




d




d

Consequently

d




Thus


=int 1

0E


dλ



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses



+int 1

0

int λ

0E

[d


]dρ dλ(211)


+int 1

0

int λ

0E

[tr


)]dρ dλ

+int 1

0

int λ

0E

[E

[tr


)]]dρ dλ




2E[E

[tr


)]](212)

+ 12E

[tr


)] + R1(PϑPϑ0) + R2(PϑPϑ0)

with the remainders

R1(PϑPϑ0) =int 1

0

int λ

0E

[E

[tr


(213)minus part2


)]]dρ dλ

and

R2(PϑPϑ0) =int 1

0

int λ

0E

[tr




dρ dλ








0gprime(E[

h(ϑ0 + sη)])


]ds

= gprime(E[h(ϑ0)

])E

[hprime(ϑ0)η

] + o(|η|L2

)= E

[gprime(E[

h(ϑ0)])

hprime(ϑ0)η] + o

(|η|L2)





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses





d



h(ϑ0)])


and


])(part2yh

)(y)


g(E

[h(ϑ)

]) minus g(E

[h(ϑ0)

])= gprime(E[

h(ϑ0)])


]+ 1


])(E


])2

+ 12gprime(E[

h(ϑ0)])

E[tr


)]+ O

((E




[Dϑf (ϑ)


[eiϑ0

]η2

Then


2D2ϑf (ϑ0)(η η)

]= E

[eiϑ0

(eiη minus [

1 + iη minus 12η2])] = E

[eiϑ0

(ei minus (1

2 + i))

IA

]= (

ei minus (12 + i

))ϕϑ0(1)P (A) = (

ei minus (12 + i

))ϕϑ0(1)|η|2

L2








dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses







dtimesd and b R




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

(31)s isin [t T ]

and

Xtxξs = x +

int s

tσ

(Xtxξ

r PX

tξr

)dBr +

int s

tb(Xtxξ

r PX

tξr

)dr

(32)s isin [t T ]


E[

supsisin[tT ]


s

∣∣2]le C




Xtxξs |x=ξ = Xtξ

s s isin [t T ](34)


XsXtxξs X

tξs

r XsXtξs

r

) = (Xtxξ

r Xtξr

) r isin [s T ](35)



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses



tξs isin



Xsηr = η +

int r

sσ

(Xsη

u PXsηu

)dBu +

int r

sb(Xsη

u PXsηu

)du r isin [s T ]

and

Xsyηr = y +

int r

sσ

(Xsyη

u PXsηu

)dBu +

int r

sb(Xsyη

u PXsηu

)du r isin [s T ]



sequently XsX

txξs η




V (t x ξ) = E[

(X

txξT P

XtξT

)]


d ξ isin L2(Ft Rd

)





E[

supsisin[tT ]


s

∣∣p]le Cp


(37)



W2(PϑPθ ) = inf(




(38)

le (E




E[

supsisin[tT ]


s

∣∣2]le CE

[|ξ1 minus ξ2|2]

(39)ξ1 ξ2 isin L2(




E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses



E[

supsisin[tv]


s

∣∣p](310)

le Cp

(|x1 minus x2|p +

int v

tW2(P

Xtξ1r

PX

tξ2r

)p dr

)



supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

le supsisin[tv]

E[∣∣Xtξ prime

1ξ1s minus X

tξ prime2ξ2

s

∣∣2] le E[

supsisin[tv]


s minus Xtξ prime

2ξ2s

∣∣2](311)

le C

(E


2∣∣2] +

int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

) v isin [t T ]








supsisin[tv]

W2(PX

tξ1s

PX

tξ2s

)2

(312)

le C

(W2(Pξ1Pξ2)

2 +int v

tW2(P

Xtξ1r

PX

tξ2r

)2 dr

)


supsisin[tT ]

W2(PX

tξ1s

PX

tξ2s


(313)t isin [0 T ] ξ1 ξ2 isin L2(

Ft Rd)


E[

supsisin[tT ]


s

∣∣p]le Cp


(314)







Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses




Ft Rd)(315)


txξs =

XtxPξs = X







11b (Rd times

P2(Rd) rarr R








P2(Rd) timesR







E[

supsisin[tT ]


txPξs minus partxX

txPξs h

∣∣2]= o




XtxPξ


partxiX

txPξ

sj = δij +dsum

k=1

int s

tpartxk

bj

(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dr

+dsum

k=1

int s

t(partxk

σj)(X

txPξr P

Xtξr

)partxi

XtxPξ

rk dBr (41)

s isin [t T ]





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses





(i) E[

supsisin[tT ]

∣∣partxXtxPξs

∣∣p]le Cp

(42)(ii) E

[sup


txPξs minus partxX

txprimePξ primes

∣∣p]le Cp



tξs

)

and b(s x) = b(xPX

tξs





tξPξs = partxX



partxXtξPξs = I +

int s

t(partxσ )

(Xtξ

r PX

tξr

)partxX

tξPξr dBr

(43)+

int s

t(partxb)

(Xtξ

r PX

tξr

)partxX



E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣p|Ft

]le sup

xisinRE

[sup


txPξs






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](45)



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses



Xtxξ+hηs = x +

int s

tσ

(X

txPξ+hηr P

Xtξ+hηr

)dBr

(46)+

int s

tb(X

txPξ+hηr P

Xtξ+hηr

)dr


Xtxξ+hηs = x +

int s

tσ

(Xtxξ+hη

r Xtξ+hηr

)dBr

(47)+

int s

tb(Xtxξ+hη

r Xtξ+hηr

)dr





s (η)|x=ξ (48)


E[

supsisin[tT ]

∣∣partxXtxP ξs

∣∣2]le C


txPξs |x=ξ

E[

supsisin[tT ]

∣∣partxXtξPξs

∣∣2|Ft

]le C P -as(49)

Using the notation


s (η)|x=ξ (410)


Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

s (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

s (η) dr

(411)+

int s

t(Dϑσ )

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ

s (η))dBr

+int s

t(Dϑ b)

(Xtxξ

r Xtξr

)(partxX

tξPξs η + Y tξ



tξr )(partxX

tξPξs η + Y

tξs (η)) we see



sure that

(Dϑσ )(yXtξ

r

)(partxX

tξPξs η + Y tξ

s (η))

(412)= E

[(partμσ)

(yP

Xtξs

Xtξs

) middot (partxX

tξPξs η + Y tξ

s (η))]

y isinR




Xtξs = ξ +

int s

tσ

(Xtξ

r PX

tξr

)dBr +

int s

tb(Xtξ

r PX

tξr

)dr

XtxPξs = x +

int s

tσ

(X

txPξr P

Xtξr

)dBr +

int s

tb(X

txPξr P

Xtξr

)dr s isin [t T ]






txPξr |x=ξ and


hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](413)


(Dϑσ )(Xtxξ

s Xtξr

) middot (partxX

tξPξs η + Y tξ

s (η))

= E[(partμσ)

(X

txPξs P

Xtξs

Xt ξ )s

) middot (partxX


s (η))]

(414)


txξs X

tξr )(partxX

tξPξs η +



Y txξs (η) =

int s

t(partxσ )

(Xtxξ

r Xtξr

)Y txξ

r (η) dBr

+int s

t(partxb)

(Xtxξ

r Xtξr

)Y txξ

r (η) dr

(415)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr



Y tξs (η) =

int s

t(partxσ )

(Xtξ

r Xtξr

)Y tξ

r (η) dBr +int s

t(partxb)

(Xtξ

r Xtξr

)Y tξ

r (η) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dBr(416)

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


r (η))]

dr


E[

supsisin[tT ]

∣∣Y txξs (η)

∣∣2]le CE

[η2]





(L2(Ft )L






hrarr0

1

h

(Xtxξ+hη

s minus Xtxξs

) s isin [t T ](418)




σ(Xtxξ+hη

s PX

tξ+hηs

) minus σ(Xtxξ

s PX

tξs

)=

int 1

0partλ

(σ

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

))dλ

(419)

+int 1

0partλ

(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))dλ

= αs(xh)(Xtxξ+hη

s minus Xtxξs

) + E[βs(xh)

(Xtξ+hη

s minus Xtξs

)]


where

αs(xh) =int 1

0(partxσ )

(Xtxξ

s + λ(Xtxξ+hη

s minus Xtxξs

)P

Xtξ+hηs

)dλ

βs(xh) =int 1

0(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))dλ



(σ

(Xtxξ

s PX

tξs +λ(X

tξ+hηs minusX

tξs )

))= (Dϑσ )

(Xtxξ

s Xtξs + λ

(Xtξ+hη

s minus Xtξs

))(Xtξ+hη

s minus Xtξs

)(420)

= E[(partμσ)

(X

txPξs P

Xtξs +λ(X

tξ+hηs minusX

tξs )

Xt ξs + λ

(Xtξ+hη

s minus Xtξs

))times (


s

)]



(X

txPξs P

Xtξs

Xt ξs

)∣∣2]le C

int 1

0

(W2(PX

tξs +λ(X

tξ+hηs minusX

tξs )

PX

tξs


s minus Xtξs

∣∣2])dλ


s minus Xtξs

∣∣2](421)

le CE[E


typrimePξs


]le CE


]le Ch2E

[η2]




(X

txPξs P

Xtξs

)∣∣p]le Cp

(E


txPξs


PX

tξs

)p)


p + C|h|pE[η2]p2

le Cp|h|pE[η2]p2




XtxPξ+hηs and X


XtxPξ+hηs minus X

txPξs =

int s

tαr(x h)

(X


)dBr

(423)+

int s

tE

[βr(xh)

(Xtξ+hη

r minus Xtξr

)]dBr


XtxPξ+hηs minus X

txPξs minus hY

txPξs (η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)(X

txPξ+hηr minus X


r (η))dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

)(424)

times (Xtξ+hη


(partxX


r (η)))]

dBr

+ R1(s x h)

with

R1(s xh)

=int s

t


(X

txPξr P

Xtξr

)) middot (X

txPξ+hηr minus X

txPξr

)dBr

+int s

tE


(X

txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]dBr



(X

txPξr P

Xtξr


txPξr

∣∣2](425)

le Ch2E[η2]


[η2]2

and (421) yields

E[∣∣E[(


txPξr P

Xtξr

Xt ξr

)) middot (Xtξ+hη

r minus Xtξr

)]∣∣2]le E

[E


txPξr P

Xtξr

Xt ξr

)∣∣2](426)


r minus Xtξr

∣∣2]]le Ch4E

[η2]2



E[

supsisin[tT ]

∣∣R1(s xh)∣∣2]

le Ch4E[η2]2




supxisinR

E[

supsisin[tu]



s (η)∣∣2]

le Ch4E[η2]2(427)

+ C

int u

tE

[E


r minus h(partxX


r (η))∣∣]2]

dr


E[∣∣Xtξ+hη


(partxX


r (η))∣∣]

= E[∣∣Xtξ+hη


(partxX

tξPξr η + Y tξ

r (η))∣∣]



r minus h(partxX

tξPξr η + Y tξ

r (η)) = I1(r h) + I2(r h)


txPξr minus hY


E[∣∣I1(r h)


E[∣∣XtxPξ+hη


r (η)∣∣2]



tξPξr η we have

E[∣∣I2(r h)

∣∣]2 le h2E[η2] int 1

0E


tξPξr

∣∣2]dλ

le h2E[η2] int 1

0E

[E


typrimePξr


]dλ

le Ch4E[η2]2


E[

supsisin[tT ]



s (η)∣∣2]

le Ch4E[η2]2




) = (Utxξ

s (y)Utξs (y)

)sisin[tT ] isin S

([t T ]R2)(428)



Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)Utxξ

r (y) dr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


(429)+ (partμσ)

(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

XtyPξr


+ (partμb)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tpartxb

(Xtξ

r PX

tξr

)Utξ

r (y) dr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


(430)+ (partμσ)

(Xtξ

r PX

tξr

Xt ξr


]dBr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

XtyPξr


+ (partμb)(Xtξ

r PX

tξr

Xt ξr


]dr



txξs = X

txPξs satisfies

Y txξs (η) = E

[Utxξ









(i) E[

supsisin[tT ]

∣∣Utxξs (y)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


s

(yprime)∣∣p]

(432)

le Cp





Utξs (y) = U



tξs (y)



E[

supsisin[tT ]

(∣∣Utxξs (y)

∣∣p + ∣∣Utξs (y)

∣∣p)] le Cp(434)


E[

supsisin[tT ]


s



(yprime)∣∣p)]

(435) le Cp



E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](436)

le Cp

(E


txprimePξ primer

∣∣2p + (E


typrimePξ primer

∣∣2])p]+ W2(PX

tξr

PX

tξ primer

)2p)12 + Cp

(E


typrimePξ primes

∣∣2])p2




E[∣∣E[

(partμσ)(X

txPξr P

Xtξr

XtyPξr


minus (partμσ)(X

txprimePξ primer P

Xtξ primer

XtyprimePξ primer


]∣∣p](437)

le Cp


and

E[∣∣E[

(partμσ)(Xtξ

r PX

tξr

XtyPξr



r PX

tξ primer

Xtξ primer


]∣∣p](438)

le Cp





E[

suprisin[ts]


r

(yprime)∣∣p]

le Cp


(439)

+ E

[int s

t


r


dr

] s isin [t T ]


E[

suprisin[ts]


r

(yprime)∣∣2]

= E[E

[sup



(yprime)∣∣2]


(440)le C

(E


[int s

tE


r

(yprime)∣∣2]

dr

]








E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


dBr(441)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ


+int s

tE

[E

[(partμb)

(Xtξ

r PX

tξr

Xt ξ Pξr


dr

+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot E[U t ξ



E[E

[(partμσ)

(Xtξ

r PX

tξr

Xt ξ Pξr


= E[(partμσ)

(Xtξ

r PX

tξr

Xt ξr



E[Utξ

s (ξ ) middot η]=

int s

tpartxσ

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tpartxb

(Xtξ

r PX

tξr

)E

[Utξ


+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX


[U t ξ


+int s

tE

[(partμb)

(Xtξ

r PX

tξr

Xt ξr

) middot (partxX

tξ Pξr middot η

+ E[U t ξ




Y tξs (η) = E

[Utξ



E[U


int s

tpartxσ

(X

txPξr P

Xtξr

)E

[U


dBr

+int s

tpartxb

(X

txPξr P

Xtξr

)E

[U


dr

(443)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dBr

+int s

tE

[(partμb)

(X

txPξr P

Xtξr

Xt ξr

) middot (partxX


r (η))]

dr


YtxPξs (η) = E

[U







txPξs isin L2(Fs)



s (η) = E[U










txξs (η) = Y txξ

s (η) = E[partμX






txξs (middot) isin



E[∣∣Y txξ

s (η)∣∣2] = E

[∣∣E[U


le E[E

[U

txPξs (ξ )2]

E[η2]] le C2E

[η2]

η isin L2(Ft )


∥∥2L(L2L2) = sup

E

[∣∣Y txξs (η)


le C


E[∣∣Y txξ


∣∣2] = E[∣∣E[(


txPξ primes (ξ )


[E


txPξ primes (ξ )

∣∣2]E

[η2]]


η isin L2(Ft )


s

∥∥2L(L2L2)

= supE












d the mapping





DξXtxξs (η) = E

[U


(E

[dsum

j=1

UtxPξ


])1leiled

(447)


((UtxPξ


UtxPξ

sij (y) =dsum

k=1

int s

tpartxk

σi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dBr

+dsum

k=1

int s

tpartxk

bi

(X

txPξr P

Xtξr

) middot UtxPξ

rkj (y) dr

+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(448)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

txPξr

dr


F(t T


Utξsij (y) =

dsumk=1

int s

tpartxk

σi

(Xtξ

r PX

tξr


r

+dsum

k=1

int s

tpartxk

bi

(Xtξ

r PX

tξr


+dsum

k=1

int s

tE

[(partμσi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

(449)+ (partμσi)k

(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dBr

+dsum

k=1

int s

tE

[(partμbi)k

(zP

Xtξr

XtyPξr

) middot partxjX

tyPξ

rk

+ (partμbi)k(zP

Xtξr

Xtξr


]∣∣∣z=X

tξr

dr






txPξ


txPξs =

(XtxPξ



partμXtxPξ


txPξs (y)

(450)= (

UtxPξ

sij (y))1leijled




d) that














partxl

(partμg(xμy)

) = partμ

(partxl


d timesP2(R

d) timesRd




)=

int 1

0E


)η]ds

=int 1

0

int 1

0E


]ds dt


] + R1((x ξ) (z η)

)





)=

int 1

0


)dt middot z

=int 1

0

int 1

0E

[partμ


)(ξ + sη)η

]ds dt middot z

= E[partμ

((partxg)(xPξ )

)(ξ)η

]z + R2

((x ξ) (z η)

)




((partxg)(xPξ )

)(y) Pξ (dy)-ae











((partxg)(xPξ )





solution of the SDE

partxXtxPξs = 1 +

int s

t(partxσ )

(X

txPξr P

Xtξr

)partxX





the SDE

part2xX

txPξs =

int s

t

((partxσ )

(X

txPξr P

Xtξr

)part2xX

txPξr

(52)+ (

part2xσ

)(X

txPξr P

Xtξr

)(partxX

txPξr




(i) E[

supsisin[tT ]

∣∣part2xX

txPξs

∣∣p]le Cp

(53)(ii) E

[sup



xXtxprimePξ primes

∣∣p]le Cp







partx

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)partx

(partμX

txPξr (y)

)dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)partμX


txPξr dBr

(54)+

int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ partx(partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]partxX

txPξr dBr

and

party

(partμXtxPξ (y)

) =int s

t(partxσ )

(X

txPξr P

Xtξr

)party

(partμX

txPξr (y)

)dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot (partxX

tyPξr

)2]dBr

(55)+

int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr



partyUtξs (y) =

int s

t(partxσ )

(Xtξ

r PX

tξr

)partyU

tξr (y) dBr

+int s

tE

[party(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)times (

partxXtyPξr

)2]dBr(56)

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr

)part2x X

tyPξr

+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)partyU

tξr (y)

]dBr s isin [t T ]




(i) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)


(partμXtxPξ (y)

)∣∣p)] le Cp

(ii) E[

supsisin[tT ]

(∣∣partx

(partμXtxPξ (y)

) minus partx


(57)+ ∣∣party

(partμXtxPξ (y)

) minus party


le Cp











E

[∣∣∣∣1

h

(X

tx+hPξs minus X

txPξs


∣∣∣∣2]le Ch2(58)



E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

((X


tx+hPξs

) minus (X

txPξ+ληs minus X

txPξs

))minus E

[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0E

[(partμX




[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[partx

(partμX


) middot η]dudv

minus E[partx

(partμX

txPξs (ξ )


le Ch2

λ2 + E

[∣∣∣∣ 1

hλ

int λ

0

int h

0E

[∣∣partx

(partμX


)minus partx

(partμX

txPξs (ξ )


∣∣∣∣2]


E[∣∣partx

(partμX


) minus partx

(partμX

txPξs (ξ )

)∣∣ middot |η|](59)


)E

[η2]12 + C|v|E[

η2]

we obtain

E

[∣∣∣∣1

λ

(partxX


txPξs

) minus E[partx

(partμX

txPξs (ξ )

) middot η]∣∣∣∣2](510)

le C

(h2

λ2 + λ2E[η2]2

)minusrarr Cλ2E

[η2]2

as h rarr 0




Dξ

(partxX

txξ )(η) = E

[partx

(partμX

txPξs (ξ )

) middot η]






Utxξs (y) =

int s

tpartxσ

(X

txPξr P

Xtξr

)Utxξ

r (y) dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr


+ (partμσ)(X

txPξr P

Xtξr

Xt ξr

)U t ξ

r (y)]dBr

Utξs (y) =

int s

tpartxσ

(Xtξ

r PX

tξr

)Utξ

r (y) dBr

+int s

tE

[(partμσ)

(Xtξ

r PX

tξr

XtyPξr


+ (partμσ)(Xtξ

r PX

tξr

Xt ξr


]dBr






hrarr0

1

h

(Utxξ+hη


) s isin [t T ]

the following SDE

Ztxξs (y η)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)Ztxξ

r (y η) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U

txPξr (y)E

[U


dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)

(partxX

tξ Pξr middot η

+ E[U t ξ


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr



dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr


]


times E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


times E[U


dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr

) middot (partx

(partμX

tξ Pξr (y)

) middot η

+ Zt ξr (y η)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr


] middot E[U


dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


(partxX

tξPξ

r middot η

+ E[U

tξ


+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+ E[U t ξ




Utξ


process Xtξ







Dϑ

[partμσ (xϑ y)

](θ) = E

[(part2μσ




tξr XtyPξ ) is

given by

Dξ

[partμσ

(X

txPξr Xtξ

r XtyPξ)]

(η)

= partx(partμσ)(X

txPξr P

Xtξr

XtyPξr

) middot E[U


+ E[(

part2μσ

)(X

txPξr P

Xtξr

XtyPξ Xtξ

r

)(514)

times (partxX

tξPξ

r middot η + E[U

tξ


(X

txPξr P

Xtξr

XtyPξr

) middot E[U


η isin L2(Ft )




UtxPξs (y z)

=int s

t(partxσ )

(X

txPξr P

Xtξr

)U

txPξr (y z) dBr

+int s

t

(part2xσ

)(X

txPξr P

Xtξr

)U


r (z) dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

XtzPξr

)U


tzPξr

]dBr

+int s

tE

[partμ(partxσ )

(X

txPξr P

Xtξr

Xt ξr

)U

txPξr (y)U tξ

r (z)]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtyPξr

) middot partμ

(partxX

tyPξr

)(z)

]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

XtyPξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tzPξ

r

)times partxX


tzPξ

r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

XtyPξr X

tξ

r


tξ

r (z)]]

dBr

(515)+

int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtyPξr


tyPξr (z)

]dBr


+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

Xt ξr


]dBr

+int s

tE

[(partμσ)

(X

txPξr P

Xtξr

XtzPξr

) middot partx

(partμX

tzPξr (y)

)]dBr

+int s

tE

[partx(partμσ)

(X

txPξr P

Xtξr

Xt ξr



+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tzPξ

r

)times U t ξ


r

]]dBr

+int s

tE

[E

[(part2μσ

)(X

txPξr P

Xtξr

Xt ξr X

tξ

r


r (z)]]

dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

XtzPξr


tzPξr

]dBr

+int s

tE

[party(partμσ)

(X

txPξr P

Xtξr

Xt ξr


r (z)]dBr








Ztxξs (y η) = E

[U





(i) E[

supsisin[tT ]

∣∣UtxPξs (y z)

∣∣p]le Cp

(ii) E[

supsisin[tT ]


txprimePξ primes


(517)

le Cp





txPξs that


tξs (y) isin


Dξ

[partμX

txPξs (y)

](η) = Dξ

[U

txPξs (y)

](η) = E

[U


Dξ

[partμXtξ

s (y)](η) = Dξ

[Utξ

s (y)](η)


[Utξ


But this means that

part2μX

txPξs (y z) = U





txPξs







(i) Dξ

[partxi

XtxPξs

](η) = E

[partμ

(partxi

XtxPξs

)(ξ ) middot η]

(ii) Dξ

[partμX

txPξs (y)

](η) = E

[part2μX


(519)




partxi

(partμX

txPξs (y)

) = partμ

(partxi

XtxPξs


and for

MtxPξs (y z)

= (part2xixj

XtxPξs partμ

(partxi

XtxPξs

)(y) part2


(partμX

txPξs (y)

))



(iii) E[

supsisin[tT ]

∣∣MtxPξs (y z)

∣∣p]le Cp

(iv) E[

supsisin[tT ]


txprimePξ primes


(520)

le Cp





d) rarr R

V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)]


d times L2(Ft Rd

)







partxiV (t xPξ ) =

dsumj=1

E[(partxj

)(X

txPξ

T PX

tξT

) middot partxiX

txPξ

T j

](62)


j=1

E[(partxj

)(X

txPξ

T PX

tξT

)(partμX

txPξ

T j

)i (y)

+ E[(partμ)j

(X

txPξ

T PX

tξT

XtyPξ

T

) middot partxiX

tyPξ

T j(63)

+ (partμ)j(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T j

)i (y)

]]



(i)∣∣partxi


∣∣ le C

(ii)∣∣partxi


(t xprimePξ prime



(64)le C



(t prime xPξ


V(t prime xPξ


(t prime xPξ y


∣∣12





tξT ) = (X

txPξ

T PX

tξT

)



txξT X

tξT )) of the



Dξ

(

(X

txξT X

tξT

))(η)

= partx(X

txξT X

tξT

)DξX

txξT (η) + (D)

(X

txξT X

tξT

)(Dξ

[X

tξT

](η)

)= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


+ E[partμ

(X

txPξ

T PX

tξT

Xt ξT

)Dξ

[X

tξT (η)

]]= partx

(X

txPξ

T PX

tξT

)E

[partμX

txPξ


[partμ

(X

txPξ

T PX

tξT

Xt ξT

) middot (partxX

tξ Pξ




txξT P

XtξT


is given by

partμ

(

(X

txPξ

T PX

tξT

))(y)

= partx(X

txPξ

T PX

tξT

)partμX

txPξ

T (y)

(66)+ E

[partμ

(X

txPξ

T PX

tξT

XtyPξ

T


T

+ partμ(X

txPξ

T PX

tξT

Xt ξT


T (y)] y isin R


E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y)

∣∣p] le Cp

E[∣∣partμ

(

(X

txPξ

T PX

tξT

))(y) minus partμ

(

(X

txprimePξ primeT P

Xtξ primeT

))(yprime)∣∣p]

(67)

le Cp




txPξ

T PX

tξT


Dξ

[V (t x ξ)

](η) = E

[Dξ

(

(X

txξT X

tξT

))(η)

]= E

[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ ) middot η]]

(68)

= E[E

[partμ

(

(X

txPξ

T PX

tξT

))(ξ )

] middot η]

that is


(

(X

txPξ

T PX

tξT

))(y)

] y isin R(69)



T PX

tξT

)] with respect


tξT







int s

0σ

(X

tξ primer+t PX

tξ primer+t

)dBt

r (610)


int s

0σ

(X

txPξ primer+t P

Xtξ primer+t

)dBt









(X

txPξ primeT P

Xtξ primeT

)](612)

= E[

(X


X0ξ primeT minust

)]



(t prime xPξ

)∣∣le E

[∣∣(X


X0ξ primeT minust

) minus (X



)∣∣](613)

le C(E



∣∣] + W2(PX

0ξ primeT minust

PX


))

le C(E





∣∣2])12




∣∣12(614)




tξ primeUtξ prime



)xyisinR

(N


))and ((


(N0ξ prime

(y)yisinR))



(

(X

txPξ primeT P

Xtξ primeT

))(y)

](615)

= E[partμ

(

(X


X0ξ primeT minust

))(y)

]



(t prime xPξ y

)∣∣le C

(E







∣∣(616)








∣∣] + W2(PX

0ξ primeT minust

PX


))



(t prime xPξ y


xyisinR(E





∣∣2(617)



∣∣2])12



(t prime xPξ y


∣∣12(618)




21b (Rd times P2(R


21b (Rd times P2(R


partxi

(partμV (t xPξ y)

) = partμ

(partxi

V (t xPξ ))(y)


d ξ isin L2(Ft Rd

)1 le i le d

and for

U(t xPξ y z

) = (part2xixj

V (t xPξ ) partxi

(partμV (t xPξ y)

)


(partμV (t xPξ y)

))




∣∣ le C



(619)le C



(t prime xPξ y z


∣∣12







partx

(partμV (t xPξ y)

) = partμ

(partxV (t xPξ )

)(y)





with


(X

txPξ

T PX

tξT

) middot (partμX

txPξ

T

)(y)

]

V2(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

XtyPξ

T


T

]]

V3(t xPξ y) = E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

]]






DξV3(t x ξ y)(η)

= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

+ partx(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

txPξ

T

](η)(622)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot Dξ

[X

tξ

T

](η)

] middot (partμX

tξ Pξ

T

)(y)

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot Dξ

[X

tξT

](η)

]]



(i) Dξ

[X

txPξ

T

](η) = E

[partμX

txPξ

T (ξ ) middot η]

(ii) Dξ

[X

tξ

T

](η) = partxX

tξPξ


tξPξ

T (ξ ) middot η]

(iii) Dξ

[X

tξT

](η) = partxX

tξ Pξ


tξ Pξ


(iv) Dξ

[(partμX

tξ Pξ

T

)(y)

](η)

= partx

(partμX

tξ Pξ


[(part2μX

tξ Pξ

T

)(y ξ ) middot η]


DξV3(t x ξ y)(η)

= E[E

[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tξ Pξ

T (y))

+ (part2μX

tξ Pξ

T

)(y ξ )

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y) middot partμX

txPξ

T (ξ )

(624)+ E

[(part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξPξ

T (ξ ))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tξ Pξ

T + partμXtξ Pξ


Therefore


= E[E

[(partμ)

(X

txPξ

T PX

tξT

Xt ξT

) middot (partx

(partμX

tzPξ

T (y)) + (

part2μX

tξ Pξ

T

)(y z)

)+ partx(partμ)

(X

txPξ

T PX

tξT

Xt ξT



T (z)

+ E[(

part2μ

)(X

txPξ

T PX

tξT

Xt ξT X

tξ

T


T (y)(625)

times (partxX

tzPξ

T + partμXtξPξ

T (z))]

+ party(partμ)(X

txPξ

T PX

tξT

Xt ξT

) middot (partμX

tξ Pξ

T

)(y)

times (partxX

tzPξ

T + partμXtξ Pξ

T (z))]]




(626)



(t xprimeP prime


(627)le C



(t prime xPξ y z


∣∣12(628)







Rd times P2(R






F(sX

txPξs P

Xtξs

) minus F(t xPξ )

=int s

t

(partrF

(rX

txPξr P

Xtξr

) +dsum

i=1

partxiF

(rX

txPξr P

Xtξr

)bi

(X

txPξr P

Xtξr

)

+ 1

2

dsumijk=1

part2xixj

F(rX

txPξr P

Xtξr

)(σikσjk)

(X

txPξr P

Xtξr

)(71)


+ E

[dsum

i=1

(partμF )i(rX

txPξr P

Xtξr

Xt ξr

)bi

(Xtξ

r PX

tξr

)

+ 1

2

dsumijk=1

partyi(partμF )j

(rX

txPξr P

Xtξr

Xt ξr

)(σikσjk)

(Xtξ

r PX

tξr

)])dr

+int s

t

dsumij=1

partxiF

(rX

txPξr P

Xtξr

)σij

(X

txPξr P

Xtξr

)dBj

r s isin [t T ]




f (PX

tξs

) minus f (Pξ )

=2nminus1sumi=0

(f (P

Xtξ

tni+1

) minus f (PX

tξ

tni

))

=2nminus1sumi=0

(E

[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)](72)

+ 1

2E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]+ 1

2E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2] + Rtni(P

Xtξ

tni+1

PX

tξ

tni

)



∣∣Rtni(P

Xtξ

tni+1

PX

tξ

tni

)∣∣ le CE

[∣∣Xtξ

tni+1minus X

tξ

tni

∣∣3]le C

(E

[(int tni+1

tni

∣∣b(X

txPξr P

Xtξr

)∣∣dr

)3](73)

+ E

[(int tni+1

tni

∣∣σ (X

txPξr P

Xtξr

)∣∣2 dr

)32])le C

(tni+1 minus tni




(i) E[partμf

(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)]

= E

[partμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

b(Xtξ

r PX

tξr

)dr

]

(ii) E[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)]]= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

σ(X

tξ

r PX

tξr

)dBr +

int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]

= E

[E

[part2μf

(P

Xtξ

tni

Xt ξ

tniX

tξ

tni

) middot(int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)

times(int tni+1

tni

b(X

tξ

r PX

tξr

)dr

)]]


(P

Xtξ

tni

Xt ξ

tni

)(X

tξ

tni+1minus X

tξ

tni

)2]= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

)(int tni+1

tni

σ(Xtξ

r PX

tξr

)dBr

+int tni+1

tni

b(Xtξ

r PX

tξr

)dr

)2]

= E

[partypartμf

(P

Xtξ

tni

Xt ξ

tni

) middotint tni+1

tni

∣∣σ (Xtξ

r PX

tξr

)∣∣2 dr

]+ Qtni


(XtxPξr P

Xtξr




f (PX

tξs

) minus f (Pξ )

=int s

tE

[b(Xtξ

r PX

tξr

)partμf

(P

Xtξr

Xt ξr

)]dr(74)

+ 1

2

int s

tE

[σ 2(

Xtξr P

Xtξr

)(partypartμf )

(P

Xtξr

Xt ξr

)]dr s isin [t T ]


tξs


partsf (PX

tξs

) = E[b(Xtξ

s PX

tξs

)partμf

(P

Xtξs

Xt ξs

)(75)

+ 12σ 2(

Xtξs P

Xtξs

)(partypartμf )

(P

Xtξs

Xt ξs

)] s isin [t T ]



tξs



tξs

) + E[b(Xtξ

s PX

tξs

)partμF

(s xP

Xtξs

Xt ξs

)+ 1

2σ 2(Xtξ

s PX

tξs

)(partypartμF )

(s xP

Xtξs

Xt ξs

)]


(sXtxPξs )(= F(sX

txPξs P

Xtξs


F(sX

txPξs P

Xtξs

) minus F(t xPξ )

= (sX

txPξs

) minus (t x)

=int s

t

(partr

(rX

txPξr

) + b(X

txPξr P

Xtξr

)partx

(rX

txPξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2x

(rX

txPξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(rX

txPξr

)dBr

=int s

t

(partrF

(rX

txPξr P

Xtξr

)(76)

+ E

[b(Xtξ

r PX

tξr

)partμF

(rX

txPξr P

Xtξr

Xt ξr

)+ 1

2σ 2(

Xtξr P

Xtξr

)(partypartμF )

(rX

txPξr P

Xtξr

Xt ξr

)]


+ b(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)+ 1

2σ 2(

XtxPξr P

Xtξr

)part2xF

(rX

txPξr P

Xtξr

))dr

+int s

tσ

(X

txPξr P

Xtξr

)partx

(X

txPξr P

Xtξr

)dBr s isin [t T ]



d timesP2(Rd)






∣∣ le C


(t xprimePξ prime


)(77)


(t prime xPξ


∣∣12




V (t xPξ ) = E[

(X

txPξ

T PX

tξT

)] = E[

(X

0xPξ

T minust PX

0ξT minust

)](78)



=int T minust

0E

[(partx

(X

0xPξr P

X0ξr

)b(X

0xPξr P

X0ξr

)+ 1

2part2x

(X

0xPξr P

X0ξr

)σ 2(

X0xPξr P

X0ξr

)(79)

+ E

[(partμ)

(X

0xPξr P

X0ξs

X0ξr

)b(X0ξ

r PX

0ξr

)+ 1

2party(partμ)

(X

0xPξr P

X0ξr

X0ξr

)σ 2(

X0ξr P

X0ξr

)])]dr




(X

0xPξ

T minust PX

0ξT minust

)b(X

0xPξ

T minust PX

0ξT minust

)+ 1

2part2x

(X

0xPξ

T minust PX

0ξT minust

)σ 2(

X0xPξ

T minust PX

0ξT minust

)(710)

+ E[(partμ)

(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)b(X

0ξT minust PX

0ξT minust

)+ 1

2party(partμ)(X

0xPξ

T minust PX

0ξT minust

X0ξT minust

)σ 2(

X0ξT minust PX

0ξT minust

)]]




d)) Then under


T PX

tξT



d))

of the PDE


i=1


+ 1

2

dsumijk=1

part2xixj


+ E

[dsum

i=1


(711)

+ 1

2

dsumijk=1


]


Ft Rd)


FRd)


(X

sXtxPξs P

Xtξs

r XsXtξs

r

) = (X

txPξr Xtξ

r

)




V (s yPϑ) = E[

(X

syPϑ

T PX

sϑT

)] = E[

(X

syPϑ

T PX

sϑT

)|Fs

](713)


V(sX

txPξs P

Xtξs

) = E[

(X

syPϑ

T PX

sϑT

)|Fs

]|(yϑ)=(X

txPξs X

tξs )

= E[

(X

sXtxPξs P

Xtξs

T PX

sXtξs

T

)|Fs

](714)

= E[

(X

txPξ

T PX

tξT

)|Fs

] s isin [t T ]


Xtξs





V(sX

txPξs P

Xtξs

) minus V (t xPξ )

=int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(715)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr

+int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

Consequently

V(sX

txPξs P

Xtξs


=int s

tpartxV

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr s isin [t T ]

and

0 =int s

t

(partrV

(rX

txPξr P

Xtξr

) + partxV(rX

txPξr P

Xtξr

)b(X

txPξr P

Xtξr

)+ 1

2part2xV

(rX

txPξr P

Xtξr

)σ 2(

XtxPξr P

Xtξr

)(717)

+ E

[partμV

(rX

txPξr P

Xtξr

Xt ξr

)b(Xtξ

r PX

tξr

)+ 1

2party(partμV )

(rX

txPξr P

Xtξr

Xt ξr

)σ 2(

Xtξr P

Xtξr

)])dr s isin [t T ]






d timesP2(Rd))


U(sX

txPξs P

Xtξs

) minus U(t xPξ )

(718)=

int s

tpartxU

(rX

txPξr P

Xtξr

)σ

(X

txPξr P

Xtξr

)dBr


U(t xPξ ) = E[U

(T X

txPξ

T PX

tξT

)|Ft

](719)

= E[

(X

txPξ

T PX

tξT

)] = V (t xPξ )


1(21)b ([0 T ] timesR



REFERENCES























R BUCKDAHN





FRANCE

AND


SHANDONG UNIVERSITY



J LI





S PENG


SHANDONG UNIVERSITY


E-MAIL pengsdueducn

C RAINER





FRANCE


Introduction

Preliminaries






Acknowledgments

References

Authors Addresses

Mean-field stochastic differential equations and ...

Documents

Transcript of Mean-field stochastic differential equations and ...