IJSA International Journal of Stochastic Analysis 2090-3340 2090-3332 Hindawi 10.1155/2017/2876961 2876961 Research Article Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations Jin Sixian 1 http://orcid.org/0000-0002-8322-9219 Schellhorn Henry 1 Stettner Lukasz Claremont Graduate University Claremont CA USA cgu.edu 2017 27 02 2017 2017 16 12 2016 01 02 2017 27 02 2017 2017 Copyright © 2017 Sixian Jin and Henry Schellhorn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

1. Introduction

In this paper we consider semilinear second-order path-dependent PDEs (PPDEs) of parabolic type. These equations were first introduced by Dupire  and Cont and Fournie  and will be defined properly in the next section.

To motivate our result, we first consider the heat equation expressed in terms of a backward time variable. For tT we look for a function v(x,t) that solves(1)vx,tt+122vx,t2x=0;(2)vx,T=Ψx.

It is well known (see, e.g., , chapter 9.2 or ) that the solution is given by the flow of the semigroup S(t); that is, v(·,t)=S(t)Ψ, where(3)StΨx=exp122x2T-tΨxi=0T-ti2ii!2iΨxx2i.

The differential operator 1/22/x2 is said to be the (infinitesimal) generator of the semigroup S(t). Consider now the path-dependent version of the heat equation:(4)Dtvxtp,t+12Dxxvxtp,t=0;vxTp,T=ΨxTp,where xtp is a continuous path on the interval [0,t] and the derivatives are Dupire’s path derivatives. Our goal is to find the generator of the semigroup (flowing the solution) of PPDEs, which we will refer to as the semigroup of the PPDE. It turns out that 1/2Dxx, that is, one-half times the second-order vertical derivative, is not the appropriate infinitesimal generator, because of path dependence. Indeed, the vertical derivative is the rate of change of the functional v(·,t) for a change at time t. The correct infinitesimal generator is equal to 1/2ωtDs2, where Ds2 is the second-order Malliavin derivative of F(ω)Ψ(xTp(ω)). An important difference is that F is now viewed as a random variable, and the (first-order) Malliavin derivative is a stochastic process in the canonical probability space for Brownian motion. The stopping path operator ωt was introduced in . Informally, the action of the stopping path operator (which we define rigorously later) is to freeze the path after time t:(5)ωtFω=Fωt,where ωt is the stopped path. The stopped Malliavin derivative ωtDs is thus an extension of both

the Dupire derivative; while the Dupire derivative corresponds to changes of the path at only one time, the iterated derivatives ωtDs1,,snn are taken with respect to changes of the canonical path at many different times s1,,sn;

the Malliavin derivative; while the Dupire derivative can be taken pathwise, as far as we know, the construction of the Malliavin derivative necessitates the introduction of a probability space.

The proof of the representation result is straightforward. Let us consider the path-independent case (1). Let B be Brownian motion. By Itô’s lemma, it is obvious that v(B(t),t) is a martingale, say Mt, and that the value of this martingale is the conditional expectation at time t of Ψ(B(T)). Consider now a general path-dependent terminal condition Ψ(B), in , Jin et al. gave a new representation of Brownian martingales Mt (with tT) as an exponential of a time-dependent generator, applied to the terminal value MTΨ(B): (6)Mt=exp12tTωtDs2dsΨB.

By the functional Feynman-Kac formula introduced in [1, 6], it is immediate that 1/2ωtD.2 is the generator of the semigroup of the PPDE.

The main advantage of the semigroup method is that the solution of the PPDE can be constructed semianalytically: indeed, the method is similar to the Cauchy-Kowalewsky method, of calculating iteratively all the Malliavin derivatives of Ψ; (6) can be rewritten indeed as(7)Mt=i=012ii!t,TiωtDsi2Ds12ΨBdsids1.

The main disadvantage can be seen immediately by considering (7): the terminal condition Ψ must be infinitely Malliavin differentiable. In contradistinction, the viscosity solution given in  necessitates Ψ to be only bounded and continuous. However, compared to the result shown in , Ψ needs only to be defined on continuous paths.

This paper is composed of two parts. In the first part, we give a rigorous proof of the result (7). Indeed, we complete the proof of Theorem 2.3 in our article ; although the statement was correct in that paper, one step of the proof was not obvious to finish. In the second part we characterize the generator of the semilinear PPDE.

2. Martingale Representation

We first introduce some basic notations of Malliavin calculus. For a detailed introduction, we refer to  and our paper . Let Ω=C([0,T],R) and (Ω,F,{Ft}t0,P) be the complete filtered probability space, where the filtration {Ft}t0 is the usual augmentation of the filtration generated by Brownian motion B on R. The canonical Brownian motion can be also denoted by Bt=Bt,ω=ωt,t0,T,ωΩ, by emphasizing its sample path. We denote by L2(P) the space of square integrable random variables. For simplicity, we denote dukdu1duk.

We denote the Malliavin derivative of order l at time t1,,tn by Dt1,,tnl. We call D([0,T]) the set of random variables which are infinitely Malliavin differentiable and FT-measurable, that is, for any integer n and FD([0,T]):(8)Esups1,,sn0,TDs1,,snnF2<+.

Definition 1.

For any deterministic function fL2([0,T]), we define the “stopping path” operator ωt for tT as(9)ωt0TfsdBs0tfsdBs.In particular, ωtB(s)=B(st) that is to “freeze” Brownian motion after time t.

From the definition, it is not hard to obtain that, for any n-variable smooth function g, ωtg(B(s1),,B(sn))=g(B(s1t),,B(snt)). For a general random variable FL2(P), ωtF refers to the value of F along the stopping scenario ωtωt(ω) of Brownian motion. According to the Wiener-Chaos decomposition, for any FL2(P), there exists a sequence of deterministic function {fn}n1 such that F=m=0Im(fm) with convergence in L2([0,T]n). Therefore, in order to obtain an explicit representation of ωt acting on a general variable F, we first show the following proposition.

Proposition 2.

Let fnL2([0,T]n), an n-variable square integrable deterministic function; then(10)Infnχ0,t=ωtInfn+k=1n/2n!2kn-2k!tu1ukTωtIn-2kfnu1,u1,,uk,ukduk.Therefore(11)ωtInfn=n!k=0n/2-1k2kn-2k!k!t,Tk0,tn-2kfns1,,sn-2k,u1,u1,,uk,ukdBsn-2kduk,as well as the isometry:(12)EωtInfn2=n!2k=0n/2122kn-2k!k!2t,Tk0,tn-2kfns1,,sn-2k,u1,u1,,uk,uk2dsn-2kduk.

Theorem 3.

Let FL2(P). Then for any fixed time t and ts<T, there exists a sequence {FN}N0 that satisfies the following:

FNF in L2(P);

DuFN=Ds+1/NFN for any us,s+1/N;

there exist ε(0,1) and a constant C which does not depend on N such that(13)EωtFN-F2CN2+ε.

We introduce the derivative d in L2(P) as, for any process Fs,(14)GsdFsdsis  defined  by  limε0EFs+ε-Fsε-Gs2=0.Then we can set up an operator differential equation for Es. The following theorem is a generalization of Theorem 2.2. in  to functionals that are not discrete.

Theorem 4.

For 0tsT, assuming that FD6([0,T]), one has(15)dωtEFFsds=-ωt12Ds2EFFs.

Then our main theorem is the integral version of this operator differential equation. We first introduce the convergence condition.

Condition 1.

For any n0, F satisfies(16)T-t2n2nn!2Esupu1,,unt,TωtDun2Du12F2n0.

According to isometry (12), this condition implies D([0,T]).

Remark 5.

We claim that other conditions exist which are easier to check than Condition 1. One of them is the convergence of the terms of series (23):(17)T-tn2nn!supu1,,unt,TωtDun2Du12Fn0a.s.To this “ local” condition, that is, a condition based on the calculation along the frozen path only, one needs to add a “global” condition involving all the paths to make it sufficient; that is, E[DsnF2]<c2n for any st,T and n1, with a constant c.

Moreover, with different structures of F, we have different alternative conditions which are easier to check for practical calculations. Here we list two examples.

If F=f0Tg(s)dB(s) with smooth deterministic function f and square integrable deterministic function g, it is not hard to obtain(18)T-tn2nn!supu1,,unt,TωtDun2Du12F=T-tn2nn!supxt,Tgx2nf2n0tgsdBs.

Therefore, if there exists a constant C such that, for all n1,(19)supxRf2nxnC2n,

with the help of Stirling approximation n!~2πnn/en, Condition 1 is satisfied.

If F has its chaos decomposition F=m=0Im(fm), we have(20)ωtDun2Du12F=m=2nm!m-2n!ωtIm-2nfm·,u1,u1,,un,un.

Then according to (12), Condition 1 can be replaced by(21)CT-t2n2nn!2m=2nm!m-2n!2supu1,,unt,T0,tm-2nfms1,,sm-2n,u1,u1,,un,un2dsm-2nn0,

with some constant C or some much stronger but easier conditions like the following: for m1(22)sups1,,sm0,Tfms1,,sm2Cm!.

Then we have the following main result.

Theorem 6.

Suppose that F satisfies Condition 1 and is FT-measurable. For tT, then, in L2(P),(23)EFFt=exp12tTωtDs2dsF.

The importance of the exponential formula (23) stems from the Dyson series representation, which we rewrite hereafter in a more convenient way:(24)EFFt=ωtF+12tTωtDs2Fds+14tTs1TωtDs12Ds22Fds2ds1+.

3. Representation of Solutions of Path-Dependent Partial Differential Equations 3.1. Functional Itô Calculus

We now introduce some key concepts of the functional Itô calculus introduced by Dupire . For more information, the reader is referred to , which we copy hereafter almost verbatim. Let T>0 be fixed. For each t0,T we denote by Λt the set of càdlàg (right continuous with left limits) R-valued functions on [0,t]. For each γtΛt, the value of γt at s0,t is denoted by γ(s). Denote Λ=t0,TΛt. For each γtΛ, Tst, and xR, we define(25)γtxrγr10,tr+γt+x1tr,r0,t,γt,srγr10,tr+γt1t,sr,r0,s.

Definition 7.

Given a function u^:ΛR, there exists pR such that (26)u^γtx=u^γt+px+oxas  x0.

Then we say that u^ is vertically differentiable at γtΛ and define Dxu^(γt)p. The function u^ is said to be vertically differentiable if Dxu^(γt) exists for each γtΛ. The second-order derivative Dxx is defined similarly.

Definition 8.

For a given γtΛ, if(27)u^γt,s=u^γt+as-t+os-tas  st,st,then we say that u^ is horizontally differentiable at γt and define Dtu^(γt)a. The function u^ is said to be horizontally differentiable if Dxu^(γt) exists for each γtΛ.

Definition 9.

The function u^ is said to be in Cl,Lip1,2(Λ) if Dtu^, Dxu^, and Dxxu^ exist and we have(28)φγt-φγ-t-C1+γtk+γ-t-kdγt,γ-t-for  each  γt,γ-t-Λ,where φ=u^,Dtu^,Dxu^,Dxxu^, C and k are some constants depending only on φ, and (29)dγt,γ-t-sups0,tt-γst-γ-st-+t-t-1/2is the distance on Λ. The classes Cl,Lip0,1 and Cl,Lip0,2 are defined analogously.

For each t0,T, we denote by Ωt the set of continuous R-valued functions on [0,t]. We denote Ω=t0,TΩt. Clearly ΩΛ. Given u^:ΛR and u:ΩR, we say that u is consistent with u^ on Ω if (since we already use the symbol ωt to denote our freezing path operator (see Definition 1), we here use ωt to denote a sample path) for each ωtΩ, (30)uωt=u^ωt.

Definition 10.

The function u:ΩR is said to be in Cl,Lip1,2(Ω) if there exists a function u^Cl,Lip1,2(Λ) such that (30) holds and for ωtΩ we denote (31)Dtuωt=Dtu^ωt,Dxuωt=Dxu^ωt,Dxxuωt=Dxxu^ωt.

Note. In the introduction, we use the notation {v(·,t)} for a family of nonanticipative functionals where v(·,t):ΛtR. In order to highlight the symmetry between PDEs and PPDEs, the notation v(xtp,t) in PPDEs shows that xtp is the counterpart of the argument x in PDEs and is used instead of ωt. This is in spirit closer to the original notation of [1, 2]. The reader will have no problem identifying u(xtp)=v(xtp,t).

3.2. Non-Markovian BSDEs

As in , we use Frt to denote the completion of the σ-algebra generated by B(s)-B(t) with st,r. Then we introduce H2(t,T), the space of all Fst-adapted R-valued processes X(s)st,T with EtTXs2ds<, and S2(t,T), the space of all Fst-adapted R-valued continuous processes X(s)st,T with Esupst,TXs2<. Denote now γγtx(r)=γ(r)10,t(r)+(γ(r)+x)1[t,T](r).

We will make the following assumptions:

(H1) Φ is a R-valued function defined on ΛT. Moreover, ΦCl,Lip1,2(ΛT).

(H2) The drift a(γt) is a given R-valued continuous function defined on Λ (see  for a definition of continuity). For any γtΛ and s0,t, the function xa((γt)γsx) is differentiable and its derivative da((γt)γsx)/dxφ(x) satisfies(32)φx-φyC1+xk+ykx-y,x,yR,where C and k are constants depending only on φ.

We now assume that (H1) and (H2) hold. We consider a non-Markovian BSDE, which is a particular case of (3.2) in . From Theorem  2.8 in , for any γtΛ, there exists a unique solution (Yγt(s),Zγt(s))tsTS2(t,T)×H2(t,T) of the following BSDE:(33)Yγts=ΦBγt+sTaBrγtYγtrdr-sTZγtrdBr,where(34)Bγtuγu10,tu+γt+Bu-Bt1t,Tu.In particular, Yγt(t) defines a deterministic mapping from Λ to R.

3.3. Path-Dependent PDEs

The drift a and terminal condition Ψ are required to be extended to the space of càdlàg paths because of the definition of the Dupire derivatives. We require the following (see  again):

(B1) The function Ψ is a R-valued function defined on ΩT. Moreover, there is a function ΦCl,Lip1,2(ΛT) such that Ψ=Φ on ΩT.

(B2) The drift a(ωt) is a given R-valued continuous function defined on Ω×R×R (see  for a definition of continuity). Moreover, there exists a function b satisfying (H2) such that a=b on Ω.

We can now define the following quasilinear parabolic path-dependent PDE:(35)Dtuωt+aωtDxuωt+12Dxxuωt=0,ωtΩ,t0,T;uωT=ΨωT,ωTΩT.

Theorem  4.2 in  states the following: let uCl,Lip1,2(Ω) be a solution of the above equation. Then we have u(ωt)=Yωt(t) for each ωtΩ, where (Yωt(s),Zωt(s))tsT is the unique solution of BSDE (33).

Theorem 11.

Suppose that, for each t0,T, the random variable(36)FexptTaBωtrdrΨBωtsatisfies Condition 1. Then the solution of (35) is (37)uωt=exp12tTωtDu2duF.

Proof.

According to (2.20) in  page 351, the solution of (33) is, for tsT,(38)Y^ωts=EexpsTaBωtrdrΦBωtFs.The result now follows by Theorem 6 and the fact that u(ωt)=Yωt(t).

We note that, in the case of no drift (a=0), we recover the result (6).

3.4. Proof of Proposition <xref ref-type="statement" rid="prop2">2</xref>

This proof is made up by several inductions. Therefore we separate them into several steps.

Step 1.

We first apply Itô’s lemma and integration by parts formula of the Skorohod integral of Brownian motion to provide an explicit expansion for In(fn). The goal of the following step is to transform Skorohod integrals into time-integrals. For example, f(s1,s2) is symmetric:(39)I2f=0T0Tfs1,s2dBs2dBs1=0TBTfs1,T-0TBs2fs2s1,s2ds2dBs1.

By the integration by parts formula (see (1.49) in ),(40)I2f=BT0Tfs1,TdBs1-0Tfs1,Tds1-0TBs20Tfs2s1,s2dBs1-0s2fs2s1,s2ds1ds2=BT2fT,T-BT0Tfs1s1,TBs1ds1-0Tfs1,Tds1-0TBs2BTfs2T,s2ds2+0T0TBs1Bs2fs1s2s1,s2ds1ds2+0T0s2fs2s1,s2ds1ds2=BT2fT,T-2BT0Tfs1s1,TBs1ds1+0T0TBs1Bs2fs1s2s1,s2ds1ds2-0Tfu,udu.

Based on this idea, for n1 and 1rn, we define(41)ArTsr+1,,snBTr+k=1r-1krkBTr-k0,Tkfns1,,sk,T,,T,sr+1,,sns1skBs1Bskdskand A0T(s1,,sn)=1. For n=0, A0T=1. Then we are going to prove (42)Infn=k=0n/2-1kn!2kn-2k!k!0,TkAn-2kTu1,u1,,uk,ukdukbased on the following recurrence formula of Ar: for any r=0,,n-1(43)0TArTsr+1,,sndBsr+1=Ar+1Tsr+2,,sn-r0TAr-1Tu,u,sr+2,,sndu.

To prove (43), we apply the integration by parts formula. For simplicity, we only keep the variables s1,,sk and sr+1. The notation x^ means that the variable x is not an argument of a function. We also emphasize again the symmetricity of function fn:(44)0TArTsr+1dBsr+1=k=0r-1krk0,Tk0Tfns1,,sk,sr+1s1skBTr-kBs1BskdBsr+1dsk=k=0r-1krk0,TkBTr-kBs1Bsk0Tfns1,,sk,sr+1s1skdBsr+1dsk-0,Tki=1kBTr-k0siBs1B^siBskBsr+1fns1,,sk,sr+1s1skdsr+1dsk-0,Tkn-kBTr-k-10TBs1Bskfns1,,sk,sr+1s1sksr+1dsr+1dsk(45)=k=0r-1krk0,TkBTr-k+1Bs1Bskfns1,,sk,Ts1skdsk(46)-0,Tk+1BTr-kBs1BskBsr+1fns1,,sk,sr+1s1sksr+1dsr+1dsk(47)-0,TkkBTr-kBs1Bsk-1fns1,,sk-1,sk,Ts1sk-1dskdsk-1(48)+0,Tk-10TkBTr-kBs1Bsk-1fns1,,sk-1,u,us1sk-1dsk-1du(49)-0,Tk+1n-kBTr-k-1Bs1Bskfns1,,sk,sr+1,Ts1skdsr+1dsk.Observing the properties of the binomial coefficients,(50)rk+1k+1-rkr-k=0;rk+rk+1=r+1k+1;rkk=rr-1k-1.We can see that, under the summation over k, (47) and (49) cancel each other, (45) and (46) combine into Ar+1T, and (48) remains as the integral of Ar-1T. Rigorously, we proved (43).

To prove (42), we use induction. Supposing that case n is correct, we observe case n+1: by (43), (51) 0 T I n f n s n + 1 d B s n + 1 = k = 0 n / 2 - 1 k n ! 2 k n - 2 k ! k ! · 0 , T k 0 T A n - 2 k T s n + 1 , u 1 , u 1 , , u k , u k d B s n + 1 d u k = k = 0 n / 2 - 1 k n ! 2 k n - 2 k ! k ! 0 , T k A n + 1 - 2 k T u 1 , u 1 , , u k , u k 0 T A n - 1 - 2 k T - n - 2 k 0 T A n - 1 - 2 k T u 1 , u 1 , , u k , u k , u k + 1 , u k + 1 d u k + 1 d u k = k = 0 n + 1 / 2 - 1 k n + 1 ! 2 k n + 1 - 2 k ! k ! · 0 , T k 0 T A n + 1 - 2 k T s n + 1 , u 1 , u 1 , , u k , u k d B s n + 1 d u k .

Step 2.

Now we are going to consider the action of the freezing path operator. We first prove that for all rn(52)ωtArTsr+1,,sn=Artsr+1,,sn.We only present the proof of r=n and the general case is the same. By definition, we know that ωtBs=Bsχ0,t(s)+Btχt,T(s). Therefore(53)ωtAnT=k=0n-1knk0,Tkfns1,,sk,T,,Ts1skBtn-kωtBs1Bskdsk=k=0n-1knk0,Tkfns1,,sk,T,,Ts1skBtn-ki=1kBsiχ0,tsi+Btχt,Tsidsk=k=0n-1knkk1=0kkk10,tk1×t,Tk-k1fns1,,sk,T,,Ts1skBtn-k1Bs1Bsk1dsk.Now we recall a basic integration rule for a smooth function gn as(54)t,Tngns1,,sns1sndsk=j=0n-1jnjgnT,,Tn-j,t,,tj.We apply (54) on (53) and obtain(55)ωtAnT=k=0nk1=0kj=0k-k1-1k+jnkkk1k-k1j0,tk1fns1sks1,,sk1,T,,Tk-k1-j,t,,tj,T,,Tn-kBtn-k1Bs1Bsk1dsk1.Since the number of variable T is n-k+k-k1-j=n-k1-j, which does not depend on k, it enlightens us to change the order of summations. We want to sum over k first. Observe that k=0nk1=0kj=0k-k1=k1=0nj=0n-k1k=j+k1n; we obtain(56)ωtAnT=k1=0nj=0n-k1k=j+k1n-1n-kn-k1-jn-k-1n-jn!k1!j!n-k1-j!0,tk1fns1sks1,,sk1,t,,tj,T,,Tn-k1-jBtn-k1Bs1Bsk1dsk1.According to the property of binomial coefficient again(57)k=j+k1n-1n-kn-k1-jn-k=0when  n>j+k1.We claim that (56) is not 0 only when n=j+k1. Thus we have(58)ωtAnT=j+k1=n-1k1nk10,tk1fns1sks1,,sk1,t,,tjBtn-k1Bs1Bsk1dsk1=Ant.

Step 3.

Now we can prove recurrence formula (10).

By (52) and (42), we have(59)Infnχ0,t=k=0n/2-1kn!2kn-2k!k!0,tkAn-2ktu1,u1,,uk,ukduk;ωtInfn=k=0n/2-1kn!2kn-2k!k!0,TkAn-2ktu1,u1,,uk,ukduk.Now we calculate the right hand side of (10):(60)k=0n/2n!2kn-2k!tu1ukTωtIn-2kfnu1,u1,,uk,ukduk=k=0n/2k1=0n/2-kt,Tk0,Tk1n!2kn-2k!-1k1n-2k!2k1n-2k-2k1!k1!An-2k-2k1tu1,u1,,uk1,uk1,v1,v1,,vk,vkduk1dvk.Let m=k+k1 and we continue the above formula:(61)=k=0n/2m=kn/2t,Tk0,Tm-k-1mn!2mn-2m!m!-1kmkAn-2mtu1,u1,,um-k,um-k,v1,v1,,vk,vkdum-kdvk=m=0n/2k=0mt,Tk0,Tm-k-1mn!2mn-2m!m!-1kmkAn-2mtu1,u1,,um-k,um-k,v1,v1,,vk,vkdum-kdvk.Now we apply another basic rule of integration, for a m-variable symmetric function gm (62) 0 , t m g m d u m = 0 , T t , T m g m d u m = k = 0 m t , T k 0 , T m - k - 1 k m k g m u 1 , , u m - k , v 1 , , v k d u m - k d v k .

Now apply (62) in (61) and we finally obtain(63)k=0n/2n!2kn-2k!tu1ukTωtIn-2kfnu1,u1,,uk,ukduk=m=0n/20,tm-1mn!2mn-2m!m!An-2mtu1,u1,,um,umdum=Infnχ0,t.

Step 4.

We now use induction to prove (11), based on (10). For simplicity, we introduce (64)an-2kt,TkωtIn-2kfnu1,u1,,uk,ukduk;bn-2kt,TkIn-2kfnχ0,tu1,u1,,uk,ukdukfor kn/2. Then (10) implies(65)an=bn-k=1n/2n!2kn-2k!k!an-2k.We calculate the right hand side of (11) with (65): let m=k+k1(66)n!k=0n/2-1k2kn-2k!k!bn-2k=n!k=0n/2k1=0n/2-k-1k2kn-2k!k!n-2k!2k1n-2k-2k1!k1!an-2k-2k1=k=0n/2k1=0n/2-k-1kn!2k+k1n-2k+k1!k1!k!an-2k-2k1=m=0n/2-1mn!2mn-2m!m!an-2mk1=0m-1k1mk1=an=ωtInfn.The proposition is proved.

3.5. Proof of Theorem <xref ref-type="statement" rid="thm3">3</xref>

The proof is constructive. For any fixed t[0,T], if F has its chaos decomposition n=0In(fn), then for fixed N (depending on M), we will study FM,Nn=0MIn(fnN), where (67)fnNs1,,snfntχs,s+1/Ns1+s1χ0,Ts,s+1/Ns1,,tχs,s+1/Nsn+snχ0,Ts,s+1/Nsn.In other words, the kernel fnN is constant when its arguments lie between s and s+1/N. Then we have the following lemma.

Lemma 12.

ω t I n ( f n N ) N L 2 ( P ) ω t I n ( f n ) and in particular(68)EωtInfnN-Infn2Cn!2n7N3,where C is a constant which does not depend on N and n.

Proof.

For any fixed n, we define a sequence of sets Ak1,k2k1+k2n as(69)Ak1,k2s1,,sn:0s1sk1tsk1+1sk1+k2t+1Nsk1+k2+1snT.Observe that on Ak1,0 the kernels fn and fnN coincide. According to (67), we obtain (70)ωtInfn-ωtInfnN=n!k1+k2n,k20ωtAk1,k2fn-fnNs1,,sndBsn.To bound (70), we apply Proposition 2 to obtain (71) E ω t I n f n 2 = n ! 2 k = 0 n 1 k ! 2 t , T k 0 s 1 s n - k t f n s 1 , , s n - k , u 1 , , u k 2 d s n - k d u k < . Now we apply (71) on (70) and by Cauchy-Schwartz inequality, we have(72)EωtInfnN-ωtInfn2nn!2k1+k2n,k20EωtAk1,k2fn-fnNs1,,sndBsn2=nn!2k1+k2n,k20k=n-k1n1k!2t,Tk0s1sn-ktfn-fnNs1,,sn-k,u1,,uk2χAk1,k2s1,,sndsn-kduk.Since fn is differentiable with respect to s1,,sn, there exists a constant Cn such that (73)fns1,x1,,sn,xn-fnt,x1,,t,xnCnnsups1,,snsi-t.Therefore following (72), we obtain(74)EωtInfnN-ωtInfn2Cnn!2n5N3,where C is a constant which does not depend on n and N.

Now we construct FN by n=0In(fnN). To prove the theorem, we introduce two subseries FM,N and FM by (75)FM,Nn=0MInfnNL2PMFN;FMn=0MInfnL2PMF.For enough large N, we choose M such that M7(M!)21/3MN. Then by Lemma 12 and Cauchy-Schwarz inequality, there exists a constant ε(0,1) such that(76)EωtFM,N-FM2=En=0MωtInfn-ωtInfnN2CMn=0Mnn!2n5N3CN2+ε.Then using triangle inequality, we prove the theorem.

3.6. Proof of Theorem <xref ref-type="statement" rid="thm4">4</xref>

For any FL2(P), st,T, we choose the sequence FNN0 constructed in Theorem 3. Then by the Clark-Ocone formula, we obtain (77)EFNFs-1/N=EFNFs-s-1/NsEDsFNFsdBs1+s-1/Nss1sEDs2FNFsdBs2dBs1-Rs-1/N,s3,where (78)Rs-1/N,s3=s-1/Nss1ss2sEDs3FNFs3dBs3dBs2dBs1.On one hand, by Lemma  5.2 in , we obtain (79)ERs-1/N,s32i=03EDs6-iFN23i4i!3!21N6-i.On the other hand, we can compute (80)ωt-s-1/NsEDsFNFsdBs1=1NωtEDs2FNFs;ωts-1/Nss1sEDs2FNFsdBs2dBs1=ωt-12NEDs2FNFs+12N2EDs4FNFs.Then we can establish the equation as(81)EωtNEFNFs-1/N-EFNFs+ωt12Ds2EFNFs22Eωt12NEDs4FNFs2+2EωtRs-1/N,s32=O1N2,where the last equality follows from (79), Proposition 2. Thus combining (77), (79), (80), and (81) as well as the assumption FD6([0,T]) and Proposition 2, we have (82)ωtNEFNFs-1/N-EFNFs-12Ds2EFNFsNL2P0.Here, for simplicity, we define L2 norm ·L2(P)2E[(·)2]. Then, from Theorem 3 and the closability of the Malliavin derivative operator, for some constant ε<1,(83)ωtEDsFNFs-ωtEDsFFsL2P2CN2+ε.With triangle inequality and Cauchy-Schwartz inequality, we finally have, using (81) and (83),(84)ωtNEFFs-1/N-EFFs-ωt12Ds2EFFsL2P2ωtNEFFs-1/N-EFNFs-1/NL2P2+ωtNEFFs-EFNFsL2P2+ωtNEFNFs-EFNFs-1/N-ωt12Ds2EFNFsL2P2+ωt12Ds2EFNFs-ωt12Ds2EFFsL2P2CNεor in other words(85)dωtEFFsds=-ωt12Ds2EFFs.

3.7. Proof of Theorem <xref ref-type="statement" rid="thm6">6</xref>

For i=1,,N(T-s), define(86)xiNNωtEFFs+i-1/N-EFFs+i/N-12NDs+1/N2EFFs+i/N.We rewrite (84) as(87)ExiNN2CN2+ε.Jensen’s inequality states that(88)i=1NExiNN2i=1NExiN2NCNε.Since sT1/2Du2EFFudu is bounded in L2(P), then (89)i=1NxiNNL2PNωtEFFs-F-sT12Du2EFFudu.Using (88), we thus proved that, in L2(P),(90)ωtEFFs=ωtF+sTωt12Ds2EFFudu.Then for positive integer n we define the operator Ts(n) by(91)TsnFi=0nAi,sF,where(92)Ai,sFss1siT12iDs12Dsi2Fdsi.Then by iterating (90) we obtain the following: for n>0(93)ωtEFFs=ωtTsn-1F+12nsu1unTωtDs12Dsn2EFFundun.Thus according to Condition 1,(94)EωtEs-Tsn-1F2=E12nsu1unTωtDs12Dsn2EFFundun2T-s2n2nn!2Esupu1,,un0,TωtDs12Dsn2F2n0.We now take s=t and obtain(95)EFFt=EtF=ωtTtF=n=012ntu1unTωtDs12Dsn2Fdun.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Dupire B. Functional Ito calculus Portfolio Research Paper 2009 2009-04 Bloomberg Cont R. Fournie D.-A. Functional Ito calculus and stochastic integral representation of martingales The Annals of Probability 2013 41 1 109 133 10.1214/11-aop721 MR3059194 2-s2.0-84874965188 McOwen R. Partial Differential Equations, Methods and Applications 1996 Prentice Hall Yosida K. Functional Analysis 1978 Springer MR0500055 Jin S. Peng Q. Schellhorn H. A representation theorem for smooth Brownian martingales Stochastics 2016 88 5 651 679 10.1080/17442508.2015.1116537 MR3484036 2-s2.0-84949780265 Peng S. Wang F. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula Science China Mathematics 2016 59 1 19 36 10.1007/s11425-015-5086-1 MR3436993 2-s2.0-84944909150 Ekren I. Keller C. Touzi N. Zhang J. On viscosity solutions of path dependent PDEs The Annals of Probability 2014 42 1 204 236 10.1214/12-aop788 MR3161485 2-s2.0-84893169031 Nualart D. The Malliavin Calculus and Related Topics 1995 Springer 10.1007/978-1-4757-2437-0 MR1344217 Yong J. Zhou X. Y. Stochastic Controls: Hamiltonian Systems and HJB Equations 1999 43 New York, NY, USA Springer Applications of Mathematics (New York) 10.1007/978-1-4612-1466-3 MR1696772