AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 194341 10.1155/2014/194341 194341 Research Article Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations http://orcid.org/0000-0002-1262-9688 Zhu Qingfeng 1, 2 http://orcid.org/0000-0001-8784-8239 Shi Yufeng 2 Yan Litan 1 School of Mathematic and Quantitative Economics Shandong University of Finance and Economics Jinan 250014 China sdufe.edu.cn 2 Institute for Financial Studies and School of Mathematics Shandong University Jinan 250199 China sdu.edu.cn 2014 2432014 2014 12 12 2013 27 01 2014 24 3 2014 2014 Copyright © 2014 Qingfeng Zhu and Yufeng Shi. 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.

Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

1. Introduction

In order to provide a probabilistic interpretation for the solutions of a class of semilinear stochastic partial differential equations (SPDEs), Pardoux and Peng  introduced the following backward doubly stochastic differential equations (BDSDEs): (1)Yt=ξ+tTf(s,Ys,Zs)ds+tTg(s,Ys,Zs)dBs-tTZsdWs,0tT. Due to their important significance to SPDEs, the researches for BDSDEs have been in the ascendant (cf.  and their references).

Peng and Shi  introduced a type of time-symmetric forward-backward stochastic differential equations, that is, the so-called fully coupled forward-backward doubly stochastic differential equations (FBDSDEs): (2)yt=x+0tf(s,ys,Ys,zs,Zs)ds+0tg(s,ys,Ys,zs,Zs)dWs-0tzsdBs,Yt=Φ(yT)+tTF(s,ys,Ys,zs,Zs)ds+tTG(s,ys,Ys,zs,Zs)dBs+tTZsdWs. In FBDSDEs (2), the forward equation is “forward” with respect to a standard stochastic integral dWt, as well as “backward” with respect to a backward stochastic integral dBt; the coupled “backward equation” is “forward” under the backward stochastic integral dBt and “backward” under the forward one. In other words, both the forward equation and the backward one are types of BDSDE (1) with different directions of stochastic integrals. Peng and Shi  proved the existence and uniqueness of solutions to FBDSDEs (2) with arbitrarily fixed time duration under some monotone assumptions. Zhu et al.  extended the results in  to different dimensional FBDSDEs and weakened the monotone assumptions. Zhu and Shi  further generalized the method of continuation by introducing the notion of bridge. FBDSDEs can provide more extensive frameworks for the probabilistic interpretation (nonlinear stochastic Feynman-Kac formula) for the solutions to a class of quasilinear SPDEs (cf. ) and stochastic Hamiltonian systems arising in stochastic optimal control problems (cf. ).

McKean-Vlasov stochastic differential equation (SDE) of the form (3)dXt=b(Xt,μt)dt+dWt,t[0,T],X0=x, where (4)b(Xt,μt)=Ωb(Xt(ω),Xt(ω))P(dω)=𝔼[b(ξ,Xt)]|ξ=Xt,b:m× being a (locally) bounded Borel measurable function and μ(t;·) being the probability distribution of the unknown process Xt, was suggested by Kac  and firstly studied by McKean . So far, numerous works have been done on the SDEs of McKean-Vlasov type and their applications; see, for example, Ahmed , Ahmed and Ding , Borkar and Kumar , Chan , Crisan and Xiong , Kotelenez , Kotelenez and Kurtz , and so on. It is worth pointing out that (3) is a particular case of the following general version: (5)Xt=x+0tb(s,Xs,𝔼ϕb[s,Xs,ξ]ξ=Xs)ds+0tσ(s,Xs,𝔼ϕσ[s,Xs,ξ]ξ=Xs)dWs, which can be regarded as a natural generalization of classical SDEs. Mathematical mean-field approaches play a crucial role in diverse areas, such as physics, chemistry, economics, finance, and games theory; see, for example, Lasry and Lions , Dawson , and Huang et al. . In a recent work of Buckdahn et al. , a notion of mean-field backward stochastic differential equations (MF-BSDEs) (6)Yt=ξ+tT𝔼f(s,ω,ω,Ys(ω),Zs(ω),Ys(ω),Zs(ω))ds-tTZsdWs, with t[0,T], was introduced to investigate one special mean-field problem in a pure stochastic approach.

Mean-field backward doubly stochastic differential equations (MF-BDSDEs) of the form (7)Yt=ξ+tTf(s,Ys,Zs,Γf(s,Ys,Zs))ds+tTg(s,Ys,Zs,Γg(s,Ys,Zs))dBs-tTZsdWs, where (8)[Γl(s,Ys,Zs)](ω)Ωθl(s,ω,ω,Ys(ω),Zs(ω),Ys(ω),mmmmmmZs(ω))P(dω), with l=f, g, were discussed by Wang et al. , Du et al. , and Xu . Under Lipschitz conditions, Du et al.  and Wang et al. , respectively, got the existence and uniqueness theorem of MF-BDSDEs. Wang et al.  gave one probabilistic interpretation for the solutions to a class of nonlocal SPDEs and the maximum principle of Pontryagin’s type for optimal control problems of MF-BDSDEs. Under locally monotone conditions, Xu  got the existence and uniqueness theorem and comparison theorem of MF-BDSDEs.

In this paper, we would like to introduce mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) of the form (9)yt=x+0t𝔼f(s,ξs)ds+0t𝔼g(s,ξs)dWs-0tzsdBs,Yt=𝔼'Φ(yT,yT)-tT𝔼F(s,ξs)ds-tT𝔼G(s,ξs)dBs-tTZsdWs, where (10)𝔼'l(s,ξs)=𝔼l(s,ys,Ys,zs,Zs,ys,Ys,zs,Zs)=Ωl(s,ω,ω,ys(ω),Ys(ω),zs(ω),Zs(ω),ys(ω),mmmmmYs(ω),zs(ω),Zs(ω))P(dω),mmmmmmmmmmmlmmmmml=f,g,F,G,𝔼'Φ(yT,yT)=ΩΦ(ω,ω,yT(ω),yT(ω))P(dω). Following the basic ideas in , we firstly discuss the existence and uniqueness of solutions for MF-FBDSDE (9), which obviously extends the results in , Wang et al. , Du et al. , and Xu . It is worth pointing out that MF-FBDSDE is not just a natural generalization of FBDSDE and MF-BDSDE from the view of mathematics. Our study on them also is motivated by the probabilistic interpretation for the solutions to some kind of nonlocal SPDEs.

As is well known to us, the research on SPDEs has increasingly been a popular issue in recent years. As one kind of them, SPDEs of the McKean-Vlasov type were discussed in . In fact, such equations were obtained as continuum limit from empirical distributions of a large number of SDEs, coupled with mean-field interaction. We also refer the readers to [21, 22] for more details along this. On the other hand, we would also like to mention the work of Buckdahn et al.  who studied one kind of nonlocal deterministic PDEs. In virtue of the “backward semigroup” method, they obtained the existence and uniqueness of viscosity solution for nonlocal PDEs via MF-BSDE (6) in a Markovian framework and McKean-Vlasov forward equations. Furthermore, Wu and Yu [32, 33] and Li and Wei  investigated PDE combined with algebra equations. Motivated by the above three cases, in this paper we will give some discussions on one kind of nonlocal SPDEs. A probabilistic interpretation for the solutions to such kind of SPDEs is derived by virtue of a connection between them and fully coupled FBDSDEs of mean-field type, which extends the results in  to the mean-field case and extends the results in [32, 33] to stochastic case.

The paper is organized as follows. In Section 2, we will present some preliminary notations needed in the whole paper. In Section 3, we consider the existence and uniqueness of solutions for MF-FBDSDE. In Section 4, we give the probabilistic interpretations for the solutions to a class of nonlocal SPDEs by means of MF-FBDSDEs.

2. Setting of the Problem

Let (Ω,,P) be a complete probability space on which are defined two mutually independent Brownian motions {Wt}t0 and {Bt}t0, with value, respectively, in d and l. We denote (11)ttWt,TB,t[0,T], where 𝒩 is the class of P-null sets of and (12)tWσ{Wr;  0rt}𝒩,t,TBσ{BT-Br;  trT}𝒩. In this case, the collection {t,t[0,T]} is neither increasing nor decreasing, while {tW,t[0,T]} is an increasing filtration and {t,TB,t[0,T]} is a decreasing filtration.

Let (Ω2,2,P2)=(Ω×Ω,,PP) be the completion of the product probability space of the above (Ω,,P) with itself, where we define t2=tt with t[0,T] and tt being the completion of t×t. It is worth noting that any random variable ξ=ξ(ω) defined on Ω can be extended naturally to Ω2 as ξ(ω,ω)=ξ(ω) with (ω,ω)Ω2. For H=n, and so forth, let L1(Ω2,2,P2;H) be the set of random variables ξΩ2H which is 2-measurable such that 𝔼2|ξ|Ω2|ξ(ω,ω)|P(dω)P(dω)<. For any ηL1(Ω2,2,P2;H), we denote (13)𝔼η(ω,·)Ωη(ω,ω)P(dω). Particularly, for example, if η1(ω,ω)=η1(ω), then (14)𝔼η1=Ωη1(ω)P(dω)=𝔼η1.

We would like to introduce some spaces of functions required in the sequel: (15)S2([0,T];n)={(sup0tT|φt|2)φ[0,T]×Ωnφt  is  t-measurablemmprocess  such  that  𝔼(sup0tT|φt|2)<},M2(0,T;n)={0Tφ[0,T]×Ωnφtis  t-measurable  processmmsuch  that  𝔼0T|φt|2dt<},L2(Ω,T,P;n)={𝔼|ξ|2<ξ:[0,T]×Ωnξ  is  T-measurable  randommmvariable  such  that  𝔼|ξ|2<}.

We will give notations as follows: (16)U=(yYzZ),U=(yYzZ),ξ=(UU),A(t,ξ)=(-Ff-Gg)(t,ξ).

Let n be the n-dimensional Euclidean space with the usual Euclidean norm |·| and the usual Euclidean inner product ·,·. The notation T appearing in the superscripts denotes the transpose of a matrix. Also, let n×l be the Hilbert space that consists of all n×l-matrices with the inner product A,B=tr{ABT}, A,Bn×d. Thus, the norm |A| of An×d is given by |A|=tr{AAT}. Let Sn be the set of all n×n symmetric matrices. All the equalities and inequalities mentioned in this paper are in the sense of dt×dP almost surely on [0,T]×Ω.

Consider the following MF-FBDSDEs: (17)yt=x+0t𝔼f(s,ξs)ds+0t𝔼g(s,ξs)dWs-0tzsdBs,Yt=𝔼'Φ(yT,yT)-tT𝔼F(s,ξs)dsYt=-tT𝔼G(s,ξs)dBs-tTZsdWs, where (18)ξs=(ys,Ys,zs,Zs,ys,Ys,zs,Zs),FΩ×[0,T]×n×n×n×l×n×dmm×n×n×n×l×n×dn,fΩ×[0,T]×n×n×n×l×n×dmm×n×n×n×l×n×dn,GΩ×[0,T]×n×n×n×l×n×dmm×n×n×n×l×n×dn×l,gΩ×[0,T]×n×n×n×l×n×dmm×n×n×n×l×n×dn×d,ΦΩ×n×nn. Note that the integral with respect to {Bt} is a “backward Itô integral," in which the integrand takes values at the right end points of the subintervals in the Riemann type sum, and the integral with respect to {Wt} is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Sokorohod integral (for details refer to ).

Definition 1.

A quadruple of t-measurable processes (y,Y,z,Z)M2(0,T;n+n+n×l+n×d) is called a solution of MF-FBDSDEs (17), if (17) is satisfied.

One assumes the following.

(H1) For each ξn+n+n×l+n×d+n+n+n×l+n×d,A(·,ξ) is an t-measurable process defined on [0,T] with A(·,0)M2(0,T;n+n+n×l+n×d+n+n+n×l+n×d).

(H2) A(t,ξ) and Φ(y) satisfy the Lipschitz conditions: there exist constants k>0 and 0<λ<1/2 such that (19)|f(t,ξ)-f(t,ξ-)|2k(|Y^|2|y^|2+|Y^|2+|z^|2+|Z^|2mmmmmmmmmmmm+|y^|2+|Y^|2+|z^|2+|Z^|2),|F(t,ξ)-F(t,ξ-)|2k(|Y^|2|y^|2+|Y^|2+|z^|2+|Z^|2mmmmmmmmmmmm+|y^|2+|Y^|2+|z^|2+|Z^|2),|g(t,ξ)-g(t,ξ-)|2k(|Y^|2|y^|2+|Y^|2+|Z^|2+|y^|2mmmmmmmmmmmm+|Y^|2+|Z^|2)+λ(|z^|2+|z^|2),|G(t,ξ)-G(t,ξ-)|2k(|Y^|2|y^|2+|Y^|2+|z^|2+|y^|2mmmmmmmmmmmm+|Y^|2+|z^|2)+λ(|Z^|2+|Z^|2),mmmmmmmξ,ξ-n+n+n×l+n×d+n+n+n×l+n×d,t[0,T],mnmy^=y-y-,Y^=Y-Y-,z^=z-z-,Z^=Z-Z-,mmmmmmmny^=y-y-,Y^=Y-Y-,mnmmmmimmmz^=z-z-,Z^=Z-Z-,|Φ(y,y)-Φ(y-,y-)|k|y-y-|+k|y-y-|,mmmmmmmmmimmnmnmmmmy,y-n.

The following monotonic conditions, introduced in , are main assumptions in this paper.

(H3) (20)𝔼A(t,ξ)-A(t,ξ-),U-U--μ|U-U-|2,hshhhhU=(y,Y,z,Z)T,U-=(y-,Y-,z-,Z-)T,hhhhlhhhhhhU=(y,Y,z,Z)T,hU-=(y-,Y-,z-,Z-)Tn×n×n×l×n×d,hhhhhhhhhhhhhhhhhhhhhhhhhhht[0,T],𝔼Φ(y,y)-Φ(y-,y-),y-y-β|y-y-|2,hlhhhhhhhhhhhhhhhhhhhhhhhhhy,y-n, where μ and β are positive constants.

3. The Unique Solvability of MF-FBDSDEs

In order to prove the existence and uniqueness result for (17) under (H1)–(H3), we need the following lemma. The lemma involves a priori estimates of solutions of the following family of MF-FBDSDEs parametrized by α[0,1]: (21)dyt=[α𝔼f(t,ξt)+f0(t)]dt-ztdBt+[α𝔼g(t,ξt)+g0(t)]dWt,dYt=[α𝔼F(t,ξt)-(1-α)μyt+F0(t)]dt+ZtdWt+[α𝔼'G(t,ξt)-(1-α)μzt+G0(t)]dBt,y0=x,YT=α𝔼Φ(yT,yT)+(1-α)yT+φ, where ξ=(y,Y,z,Z,y,Y,z,Z), and (F0,f0,G0,g0)M2(0,T;Rn+n+n×l+n×d), and φL2(Ω,T, P;Rn) are arbitrarily given vector-valued random variables.

When α=1, the existence of the solution of (21) implies clearly that of (17). Due to the existence and uniqueness of MF-BDSDE , when α=0, (21) is uniquely solvable. The following a priori lemma is a key step in the proof of the method of continuation. It shows that, for a fixed α=α0[0,1), if (21) is uniquely solvable, then it is also uniquely solvable for any α[α0,α0+δ0], for some positive constant δ0 independent of α0.

Lemma 2.

Under assumptions (H1)–(H3), there exists a positive constant δ0 such that if, a priori, for some α0[0,1) and for each xn,φL2(Ω,T,P;n), (F0,f0,G0,g0)M2(0,T;n+n+n×l+n×d), (21) has a unique solution, then, for each α[α0,α0+δ0] and xn, φL2(Ω,T,P;n), (F0,f0,G0, g0)M2(0,T;n+n+n×l+n×d), (21) also has a unique solution in M2(0,T;n+n+n×l+n×d).

Proof.

Let (22)U=(y,Y,z,Z),U~=(y~,Y~,z~,Z~),U-=(y-,Y-,z-,Z-),U~-=(y~-,Y~-,z~-,Z~-),ξ=(y,Y,z,Z,y,Y,z,Z),ξ~=(y~,Y~,z~,Z~,y~,Y~,z~,Z~),ξ-=(y-,Y-,z-,Z-,y-,Y-,z-,Z-),ξ~-=(y~-,Y~-,z~-,Z~-,y~-,Y~-,z~-,Z~-),ξ^=ξ-ξ~,ξ-^=ξ--ξ-~,U^=(y^,Y^,z^,Z^)=(y-y~,Y-Y~,z-z~,Z-Z~),U-^=(y-^,Y-^,z-^,Z-^)=(y--y~-,Y--Y~-,z--z~-,Z--Z~-). Since for any xn, (F0,f0,G0,g0)M2(0,T;n+n+n×l+n×d), φL2(Ω,T,P;n), there exists a unique solution to (21) for α=α0, thus, for each U-=(y-,Y-,z-,Z-)M2(0,T; n+n+n×l+n×d), there exists a unique quadruple U=(y,Y,z,Z)M2(0,T;n+n+n×l+n×d) satisfying the following equations: (23)dyt=[α0𝔼f(t,ξt)+δ𝔼f(t,ξ-t)+f0(t)]dt-ztdBt+[α0𝔼'g(t,ξt)+δ𝔼g(t,ξ-t)+g0(t)]dWt,dYt=[α0𝔼F(t,ξt)-(1-α0)μytm+δ(𝔼F(t,ξ-t)+μy-t)+F0(t)]dt+ZtdWt+[α0𝔼G(t,ξt)-(1-α0)μztml+δ(𝔼G(t,ξ-t)+μz-t)+G0(t)]dBt,y0=x,YT=α0𝔼'Φ(yT,yT)+(1-α0)yT+δ(𝔼'Φ(y-T,y-T)-y-T)+φ, where δ is a positive number independent of α0 and less than 1. We will prove that the mapping defined by (24)U=Iα0+δ(U-):M2(0,T;n+n+n×l+n×d)M2(0,T;n+n+n×l+n×d) is contractive for a small enough δ. Let U~-=(y~-,Y~-,z~-,Z~-)M2(0,T;n+n+n×l+n×d) and U~=(y~,Y~,z~,Z~)=Iα0+δ(U~-).

Applying Itô’s formula to y^,Y^ on [0,T], it follows that (25)𝔼y^T,α0Φ^(yT)+(1-α0)y^T-𝔼0T𝔼'α0(A(t,ξt)-A(t,ξ~t)),U^tdt+(1-α0)μ𝔼0T(|y^t|2+|z^t|2)dt=𝔼y^T,δy-^T-𝔼y^T,δΦ^(y-T)+δ𝔼0T(Y^t,f^(t,ξ-t)+y^t,F^(t,ξ-t)+Z^t,g^(t,ξ-t)+z^t,G^(t,ξ-t))dt+δμ𝔼0T(y^t,y-^t+z^t,z-^t)dt, where (26)f^(t,ξ-t)=𝔼'f(t,ξ-t)-𝔼'f(t,ξ~-t),g^(t,ξ-t)=𝔼'g(t,ξ-t)-𝔼'g(t,ξ~-t),F^(t,ξ-t)=𝔼'F(t,ξ-t)-𝔼'F(t,ξ~-t),G^(t,ξ-t)=𝔼'G(t,ξ-t)-𝔼'G(t,ξ~-t),Φ^(y-T)=𝔼Φ(y-T,y-T)-𝔼Φ(y~-T,y~-T),Φ^(yT)=𝔼'Φ(yT,yT)-𝔼'Φ(y~Ty~T). By virtue of (H1)–(H3), we easily deduce (27)(1-α0+α0β)𝔼|y^T|2+μ𝔼0T(|y^t|2+|z^t|2)dtδC𝔼0T(|U^t|2+|U-^t|2)dt+δC(𝔼|y^T|2+𝔼|y-^T|2), with some constant C>0. Hereafter, C will be some generic constant, which can be different from line to line and depends only on the Lipschitz constants k, μ, λ, and β. It is obvious that 1-α0+α0ββ-, β-=min(1,β)>0.

On the other hand, for the difference of the solutions (Y^,Z^)=(Y-Y~,Z-Z~), we apply a standard method of estimation. Applying Itô’s formula to |Y^t|2 on [t,T], we have (28)𝔼|Y^t|2+𝔼tT|Z^s|2ds=𝔼|α0Φ^(yT)+(1-α0)y^T+δ(Φ^(y-T)-y-^T)|2-2𝔼tTY^s,α0F^(s,ξs)-(1-α0)μy^smmmmm+δ(F^(s,ξ-s)+μy-^s)ds+𝔼tT|α0G^(s,ξs)-(1-α0)μz^s+δ(G^(s,ξ-s)+μz-^s)|2ds, where (29)F^(t,ξt)=𝔼'F(t,ξt)-𝔼'F(t,ξ~t),F^(t,ξ-t)=𝔼'F(t,ξ-t)-𝔼'F(t,ξ~-t),G^(t,ξt)=𝔼'G(t,ξt)-𝔼'G(t,ξ~t),G^(t,ξ-t)=𝔼'G(t,ξ-t)-𝔼'G(t,ξ~-t),Φ^(yT)=𝔼'Φ(yT,yT)-𝔼'Φ(y~Ty~T),Φ^(y-T)=𝔼'Φ(y-T,y-T)-𝔼'Φ(y~-T,y~-T). By virtue of (H2), we have (30)𝔼|Y^t|2+𝔼tT|Z^s|2ds4𝔼[α02|Φ^(yT)|2+(1-α0)2|y^T|2mmm+δ2|Φ^(y-T)|2+δ2|y-^T|2]+2𝔼tT|Y^s|(α0|F^(s,ξs)|+(1-α0)μ|y^s|mmmmmmmm+δ|F^(s,ξ-s)|+δμ|y-^s|)ds+𝔼tT[1+2λ4λα02|G^(s,ξs)|2]ds+3𝔼tT[1+2λ1-2λ(|G^(s,ξ-s)|2(1-α0)2μ2|z^s|2mmmmmmmmmm+δ2|G^(s,ξ-s)|2+δ2μ2|z-^s|2)1+2λ1-2λ]dsC𝔼|y^T|2+δC𝔼|y-^T|2+𝔼tT[(8k1-2λ|Y^s|2+1-2λ8k|F^(s,ξs)|2)mmmm+(1-α0)μ(|Y^s|2+|y^s|2)8k1-2λ|Y^s|2]ds+δ𝔼tT[(|Y^s|2+|F^(s,ξ-s)|2)+μ(|Y^s|2+|y-^s|2)]ds+𝔼tT[1+2λ4λ|G^(s,ξs)|2]ds+3𝔼tT[1+2λ1-2λ(|G^(s,ξ-s)|2|G^(s,ξ-s)|2(1-α0)2μ2|z^s|2mmmmmmmmmm+δ2|G^(s,ξ-s)|2+δ2μ2|z-^s|2)1+2λ1-2λ]dsC𝔼|y^T|2+δC𝔼|y-^T|2+C𝔼tT|Y^s|2ds+δC𝔼tT|ξ-^s|2ds+C𝔼tT(|y^s|2+|z^s|2)ds+3+2λ4𝔼tT|Z^s|2ds. Thus, we have (31)𝔼|Y^t|2+1-2λ4𝔼tT|Z^s|2dsC𝔼tT|Y^s|2ds+C(𝔼|y^T|2+δ𝔼|y-^T|2)+C𝔼tT(|y^s|2+|z^s|2+δ|U-^s|2)ds. By Gronwall’s inequality, it follows that (32)𝔼|Y^t|2+𝔼tT|Z^s|2dsC(𝔼|y^T|2+δ𝔼|y-^T|2)+C𝔼0T(|y^t|2+|z^t|2+δ|U-^t|2)dt. Then, we can deduce (33)𝔼0T(|Y^t|2+|Z^t|2)dtC(𝔼|y^T|2+δ𝔼|y-^T|2)+C𝔼0T(|y^t|2+|z^t|2+δ|U-^t|2)dt. Combining the above two estimates (27) and (33), for a sufficiently large constant C>0, we easily have (34)𝔼0T|U^t|2dt+𝔼|y^T|2δC(𝔼0T|U^t|2dt+𝔼|y^T|2+𝔼0T|U-^t|2dt+𝔼|y-^T|2). We now choose δ0=1/3C. It is clear that, for each fixed δ[0,δ0], the mapping Iα0+δ is contractive in the sense that (35)𝔼0T|U^t|2dt+𝔼|y^T|212(𝔼0T|U-^t|2dt+𝔼|y-^T|2). Thus, this mapping has a unique fixed point U=(y,Y,z,Z)M2(0,T;n+n+n×l+n×d), which is the solution of (21) for α=α0+δ, as δ[0,δ0]. The proof is complete.

Now we can obtain one of the main results in this paper which is the following existence and uniqueness theorem for solutions of MF-FBDSDE (17).

Theorem 3.

Under assumptions (H1)–(H3), (17) has a unique solution in M2(0,T;n+n+n×l+n×d).

Proof.

Uniqueness: let U=(y,Y,z,Z) and U-=(y-,Y-,z-,Z-) be two solutions of (17). We use the same notations as in Lemma 2. Applying Itô’s formula to y^,Y^ on [0,T], we have (36)𝔼y^T,Φ^(yT)=𝔼0T𝔼A(t,Ut)-A(t,U-t),U^tdt. By virtue of (H3), it follows that (37)μ𝔼0T𝔼A(t,Ut)-A(t,U-t),U^tdt0. Thus, UU. The uniqueness is proven.

Existence: when α=0, (21) has a unique solution in M2(0,T;n+n+n×l+n×d). It follows from Lemma 2 that there exists a positive constant δ0=δ0(k,λ,μ,β) such that, for any δ[0,δ0] and xn, φL2(Ω,T,P;n), and (F0,f0,G0,g0)M2(0,T;n+n+n×l+n×d), (21) has a unique solution for α=δ. Since δ0 depends only on (k,λ,μ,β), we can repeat this process for N times with 1Nδ0<1+δ0. In particular, for α=1 with (F0,f0,G0,g0)0 and φ0, (17) has a unique solution in M2(0,T;n+n+n×l+n×d). The proof is complete.

Remark 4.

Condition (H3) can be replaced by the following condition.

(H3)′(38)mmn𝔼A(t,ξ)-A(t,ξ-),U-U-μ|U-U-|2,mmnmiU=(y,Y,z,Z)T,U-=(y-,Y-,z-,Z-)T,mmmnnmmmmmnmmmU=(y,Y,z,Z)T,mU-=(y-,Y-,z-,Z-)Tn×n×n×l×n×d,mmmmmmmmmmmmmmmmmmit[0,T].𝔼'Φ(y,y)-Φ(y-,y-),y-y--β|y-y-|2,mmmmmmmmmmmmmmllmmmmy,y-n, where μ and β are positive constants.

By similar arguments to Theorem 3, we have another parallel existence and uniqueness theorem for MF-FBDSDEs.

Theorem 5.

Under assumptions (H1), (H2), and (H3), MF-FBDSDE (17) has a unique solution in M2(0,T;Rn+n+n×l+n×d).

4. Probabilistic Interpretation for a Class of Nonlocal SPDEs

The connection of BDSDEs and systems of second-order quasilinear SPDEs was observed by Pardoux and Peng . This can be regarded as a stochastic version of the well-known Feynman-Kac formula which gives a probabilistic interpretation for second-order SPDEs of parabolic types. Thereafter this subject has attracted many mathematicians; refer to Bally and Matoussi , Gomez et al. , Hu and Ren , Ren et al. ; see also Zhang and Zhao . One distinctive character of this result is that the forward component of the MF-FBDSDE is coupled with the backward variable. This section can be viewed as a continuation of such a theme and will exploit the above theory of fully coupled MF-FBDSDE in order to provide a probabilistic formula for the solution of a quasilinear nonlocal SPDE combined with algebra equations.

For each xn, consider the following MF-FBDSDE: (39)dys=𝔼'f(s,ξs)ds+𝔼'g(s,ξs)dWs-zsdBs,dYs=𝔼'F(s,ξs)ds+𝔼'G(s,ξs)dBs+ZsdWs,yt=x,YT=𝔼'Φ(yT,yT), where (40)ξs=(ys,Ys,zs,Zs,ys,Ys,zs,Zs),F[t,T]×n×n×n×l×n×d×nmn×n×n×l×n×dn,f[t,T]×n×n×n×l×n×d×nmn×n×n×l×n×dn,G[t,T]×n×n×n×l×n×d×nmn×n×n×l×n×dn×l,g[t,T]×n×n×n×l×n×d×nmn×n×n×l×n×dn×d,Φn×nn. Assume that (F,f,G,g,Φ) in MF-FBDSDE (39) are deterministic, and MF-FBDSDE (39) has a unique measurable solution (ys,Ys,zs,Zs), s[t,T]. Set (41)u(t,x)=Ytt,x,v(t,x)=Ztt,x. By the uniqueness of the solution to (39), it is known that, for any tsT, (42)Yst,x=Yss,yst,x=u(s,yst,x).

To simplify the notation, for φ=F,f,G,g, we define (43)φ^(s,ys0,x0,x)𝔼[φ(s,ys0,x0,x,u(s,ys0,x0),u(s,x),μ(s,ys0,x0),μ(s,x),v(s,ys0,x0),v(s,x))]. According to our notations introduced in Section 2, we know that (44)φ^(s,ys0,x0,x)=𝔼'[φ(ω,s,y0,x0(ω,s),x,u(s,y0,x0(ω,s)),u(s,x),μ(s,y0,x0(ω,s)),μ(s,x),v(s,y0,x0(ω,s)),v(s,x))].

If there exists u(t,x)C1,2(Ω×[0,T]×n;n) solving the following quasilinear second-order nonlocal SPDE: (45)u(t,x)=𝔼[Φ(yT0,x0,x)]mmmmm+tT[u(s,x)+F^(s,ys0,x0,x)]dsmmmmm+tTpG^(s,ys0,x0,x)dBs,u(t,x)μ(t,x)=qG^(t,yt0,x0,x),p+q=1,mmmmmmmmmmimmmlnmmnq0,p,q,v(t,x)=u(t,x)g^(t,yt0,x0,x),(t,x)[0,T]×n, where u:+×nm, (46)u=(Lu1Lum), with (47)Luk(t,x):=i=1nf^i(t,yt0,x0,x)ukxi(t,x)+12i,j=1n𝔼(g^g^T)ij(t,yt0,x0,x)2ukxixj(t,x)-12i,j=1n2ukxixj(t,x)(μ(t,x)μ(t,x)T)ij,mmmmmmmmmmmmmk=1,,m, we can assert the following.

Theorem 6.

Assume that (F,f,G,g,Φ) in MF-FBDSDE (39) are deterministic, and MF-FBDSDE (39) admits a unique measurable solution, the functions F, f, G, and g are of class C3. and Φ is of class C2. If (u,v) solves nonlocal SPDE (45), then (41) holds, where (Y,Z) is determined uniquely by (39).

Proof.

It suffices to show that {u(s,yst,x),g^(s,ys0,x0,x)u(s,yst,x);0st} solves MF-FBDSDE (39).

Let t=t0<t1<t2<<tn=T; we have (48)u(ti,ytit,x)-u(ti+1,yti+1t,x)=u(ti,ytit,x)-u(ti,yti+1t,x)+u(ti,yti+1t,x)-u(ti+1,yti+1t,x)=-titi+1u(ti,yst,x)ds+titi+1u(ti,yst,x)zsdBs+titi+1g^(ti,ys0,x0,yst,x)u(s,yst,x)dWs+titi+1[u(s,yti+1t,x)+F^(s,ys0,x0,yti+1t,x)]ds+titi+1pG^(s,ys0,x0,yti+1t,x)dBs, where we have used Itô's formula and the equation satisfied by u. Finally, let the mesh size go to zero; we have (49)u(t,yt)-u(T,yT)=tTF^(s,ys0,x0,yst,x)ds+tTG^(s,ys0,x0,yst,x)dBs+tTg^(s,ys0,x0,yst,x)u(s,yst,x)dWs. It is easy to check that Yst,x:=u(s,yst,x),Zst,x:=g^(s,ys0,x0,x)u(s,yst,x) coincides with the unique solution to MF-BDSDE of (39).

Remark 7.

In the case when p=0 in nonlocal SPDE (45), nonlocal SPDE (45) will degenerate to the following nonlocal PDE: (50)u(t,x)=𝔼[Φ(yT0,x0,x)]mmmmm+tT[u(s,x)+F^(s,ys0,x0,x)]ds,u(t,x)μ(t,x)=G^(t,yt0,x0,x),v(t,x)=u(t,x)g^(t,yt0,x0,x),(t,x)[0,T]×n.

Equation (41) can be called a Feynman-Kac formula for nonlocal SPDE (45).

Equation (41) generalizes the PDE combined with algebra equations in [32, 33] to the mean-field case.

By virtue of a connection between them and fully coupled FBDSDE of mean-field type, Theorem 6 gives a probabilistic interpretation for the solutions to such kind of SPDE (45). Furthermore, the uniqueness for SPDE (45) is an interesting problem, and we hope to be able to address this issue in our future publications.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referees and the editors for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant nos. 11371226, 11071145, 11301298, 11201268, and 11231005), Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant no. 11221061), the 111 Project (Grant no. B12023), and the Natural Science Foundation of Shandong Province of China (Grant no. ZR2012AQ013).

Pardoux É. Peng S. G. Backward doubly stochastic differential equations and systems of quasilinear SPDEs Probability Theory and Related Fields 1994 98 2 209 227 10.1007/BF01192514 MR1258986 Bally V. Matoussi A. Weak solutions for SPDEs and backward doubly stochastic differential equations Journal of Theoretical Probability 2001 14 1 125 164 10.1023/A:1007825232513 MR1822898 Gomez A. Lee K. Mueller C. Wei A. Xiong J. Strong uniqueness for an SPDE via backward doubly stochastic differential equations Statistics & Probability Letters 2013 83 10 2186 2190 10.1016/j.spl.2013.06.010 MR3093800 Hu L. Y. Ren Y. Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes Journal of Computational and Applied Mathematics 2009 229 1 230 239 10.1016/j.cam.2008.10.027 MR2522516 Ren Y. Lin A. Hu L. Y. Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes Journal of Computational and Applied Mathematics 2009 223 2 901 907 10.1016/j.cam.2008.03.008 MR2478888 Zhang Q. Zhao H. Z. Stationary solutions of SPDEs and infinite horizon BDSDEs Journal of Functional Analysis 2007 252 1 171 219 10.1016/j.jfa.2007.06.019 MR2357354 Zhang Q. Zhao H. Z. Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients Journal of Differential Equations 2010 248 5 953 991 10.1016/j.jde.2009.12.013 MR2592878 Zhang Q. Zhao H. Z. SPDEs with polynomial growth coefficients and the Malliavin calculus method Stochastic Processes and Their Applications 2013 123 6 2228 2271 10.1016/j.spa.2013.02.004 MR3038503 Peng S. G. Shi Y. F. A type of time-symmetric forward-backward stochastic differential equations Comptes Rendus Mathematique 2003 336 9 773 778 10.1016/S1631-073X(03)00183-3 MR1989279 Zhu Q. F. Shi Y. F. Gong X. J. Solutions to general forward-backward doubly stochastic differential equations Applied Mathematics and Mechanics 2009 30 4 517 526 10.1007/s10483-009-0412-x MR2513431 Zhu Q. F. Shi Y. F. Forward-backward doubly stochastic differential equations and related stochastic partial differential equations Science China Mathematics 2012 55 12 2517 2534 10.1007/s11425-012-4411-1 MR3001067 Han Y. C. Peng S. G. Wu Z. Maximum principle for backward doubly stochastic control systems with applications SIAM Journal on Control and Optimization 2010 48 7 4224 4241 10.1137/080743561 MR2665464 Zhang L. Q. Shi Y. F. Maximum principle for forward-backward doubly stochastic control systems and applications ESIAM: Control, Optimisation and Calculus of Variations 2011 17 4 1174 1197 10.1051/cocv/2010042 MR2859871 Shi Y. F. Zhu Q. F. Partially observed optimal controls of forward-backward doubly stochastic systems ESIAM: Control, Optimisation and Calculus of Variations 2013 19 3 828 843 10.1051/cocv/2012035 MR3092364 Kac M. Foundations of kinetic theory 3 Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability 1956 Berkeley, Calif, USA 171 197 MR0084985 McKean, H. P. Jr. A class of Markov processes associated with nonlinear parabolic equations Proceedings of the National Academy of Sciences of the United States of America 1966 56 6 1907 1911 MR0221595 Ahmed N. Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control SIAM Journal on Control and Optimization 2007 46 1 356 378 10.1137/050645944 MR2299633 Ahmed N. Ding X. A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space Stochastic Processes and Their Applications 1995 60 1 65 85 10.1016/0304-4149(95)00050-X MR1362319 Borkar V. Kumar K. McKean-Vlasov limit in portfolio optimization Stochastic Analysis and Applications 2010 28 5 884 906 10.1080/07362994.2010.482836 MR2739322 Chan T. Dynamics of the McKean-Vlasov equation The Annals of Probability 1994 22 1 431 441 MR1258884 Crisan D. Xiong J. Approximate McKean-Vlasov representations for a class of SPDEs Stochastics 2010 82 1–3 53 68 10.1080/17442500902723575 MR2677539 Kotelenez P. A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation Probability Theory and Related Fields 1995 102 2 159 188 10.1007/BF01213387 MR1337250 Kotelenez P. Kurtz T. Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type Probability Theory and Related Fields 2010 146 1-2 189 222 10.1007/s00440-008-0188-0 MR2550362 Lasry J. Lions P. Mean field games Japanese Journal of Mathematics 2007 2 1 229 260 10.1007/s11537-007-0657-8 MR2295621 Dawson D. Critical dynamics and fluctuations for a mean-field model of cooperative behavior Journal of Statistical Physics 1983 31 1 29 85 10.1007/BF01010922 MR711469 Huang M. Y. Malhamé R. Caines P. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle Communications in Information and Systems 2006 6 3 221 252 MR2346927 Buckdahn R. Djehiche B. Li J. Peng S. Mean-field backward stochastic differential equations: a limit approach The Annals of Probability 2009 37 4 1524 1565 10.1214/08-AOP442 MR2546754 Wang T. X. Shi Y. F. Zhu Q. F. Mean-field backward doubly stochastic differential equations and applications http://arxiv.org/abs/1108.5590 Du H. Peng Y. Wang Y. Mean-field backward doubly stochastic differential equations and its applications Proceedings of the 31st Chinese Control Conference (CCC '12) 2012 Hefei, China 1547 1552 Xu R. M. Mean-field backward doubly stochastic differential equations and related SPDEs Boundary Value Problems 2012 2012, article 114 10.1186/1687-2770-2012-114 MR3016038 Buckdahn R. Li J. Peng S. G. Mean-field backward stochastic differential equations and related partial differential equations Stochastic Processes and Their Applications 2009 119 10 3133 3154 10.1016/j.spa.2009.05.002 MR2568268 Wu Z. Yu Z. Y. Fully coupled forward-backward stochastic differential equations and related partial differential equations system Chinese Annals of Mathematics A 2004 25 4 457 468 MR2100490 Wu Z. Yu Z. Y. Probabilistic interpretation for systems of parabolic partial differential equations combined with algebra equations Li J. Wei Q. M. Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations http://arxiv.org/abs/1302.0935 Nualart D. Pardoux É. Stochastic calculus with anticipating integrands Probability Theory and Related Fields 1988 78 4 535 581 10.1007/BF00353876 MR950346