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Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations
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.
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): Due to their important significance to SPDEs, the researches for BDSDEs have been in the ascendant (cf. [2–8] 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): In FBDSDEs (2), the forward equation is “forward” with respect to a standard stochastic integral , as well as “backward” with respect to a backward stochastic integral ; the coupled “backward equation” is “forward” under the backward stochastic integral 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. [12–14]).
McKean-Vlasov stochastic differential equation (SDE) of the form where being a (locally) bounded Borel measurable function and being the probability distribution of the unknown process , 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: 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) with , 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 where with , , 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 where 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 be a complete probability space on which are defined two mutually independent Brownian motions and , with value, respectively, in and . We denote where is the class of -null sets of and In this case, the collection is neither increasing nor decreasing, while is an increasing filtration and is a decreasing filtration.
Let be the completion of the product probability space of the above with itself, where we define with and being the completion of . It is worth noting that any random variable defined on can be extended naturally to as with . For , and so forth, let be the set of random variables which is -measurable such that . For any , we denote Particularly, for example, if , then
We would like to introduce some spaces of functions required in the sequel:
We will give notations as follows:
Let be the -dimensional Euclidean space with the usual Euclidean norm and the usual Euclidean inner product . The notation appearing in the superscripts denotes the transpose of a matrix. Also, let be the Hilbert space that consists of all -matrices with the inner product , . Thus, the norm of is given by . Let be the set of all symmetric matrices. All the equalities and inequalities mentioned in this paper are in the sense of almost surely on .
Consider the following MF-FBDSDEs: where Note that the integral with respect to 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 is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Sokorohod integral (for details refer to ).
One assumes the following.
(H1) For each is an measurable process defined on with .
(H2) and satisfy the Lipschitz conditions: there exist constants and such that
The following monotonic conditions, introduced in , are main assumptions in this paper.
(H3) where and 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 : where , and , and , are arbitrarily given vector-valued random variables.
When , the existence of the solution of (21) implies clearly that of (17). Due to the existence and uniqueness of MF-BDSDE , when , (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 , if (21) is uniquely solvable, then it is also uniquely solvable for any , for some positive constant independent of .
Lemma 2. Under assumptions (H1)–(H3), there exists a positive constant such that if, a priori, for some and for each ,, , (21) has a unique solution, then, for each and , , , , (21) also has a unique solution in .
Since for any , , , there exists a unique solution to (21) for , thus, for each ; , there exists a unique quadruple satisfying the following equations:
where is a positive number independent of and less than . We will prove that the mapping defined by
is contractive for a small enough . Let and .
Applying Itô’s formula to on , it follows that where By virtue of (H1)–(H3), we easily deduce with some constant . Hereafter, will be some generic constant, which can be different from line to line and depends only on the Lipschitz constants , , , and . It is obvious that , .
On the other hand, for the difference of the solutions , we apply a standard method of estimation. Applying Itô’s formula to on , we have where By virtue of (H2), we have Thus, we have By Gronwall’s inequality, it follows that Then, we can deduce Combining the above two estimates (27) and (33), for a sufficiently large constant , we easily have We now choose . It is clear that, for each fixed , the mapping is contractive in the sense that Thus, this mapping has a unique fixed point , which is the solution of (21) for , as . 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 .
Proof. Uniqueness: let and be two solutions of (17). We use the same notations as in Lemma 2. Applying Itô’s formula to on , we have
By virtue of (H3), it follows that
Thus, . The uniqueness is proven.
Existence: when , (21) has a unique solution in . It follows from Lemma 2 that there exists a positive constant such that, for any and , , and , (21) has a unique solution for . Since depends only on , we can repeat this process for times with . In particular, for with and , (17) has a unique solution in . The proof is complete.
Remark 4. Condition (H3) can be replaced by the following condition.
(H3)′ where and 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 .
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 [6–8]. 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 , consider the following MF-FBDSDE: where Assume that in MF-FBDSDE (39) are deterministic, and MF-FBDSDE (39) has a unique measurable solution , . Set By the uniqueness of the solution to (39), it is known that, for any ,
To simplify the notation, for , we define According to our notations introduced in Section 2, we know that
If there exists solving the following quasilinear second-order nonlocal SPDE: where , with we can assert the following.
Theorem 6. Assume that in MF-FBDSDE (39) are deterministic, and MF-FBDSDE (39) admits a unique measurable solution, the functions , , , and are of class . and is of class . If solves nonlocal SPDE (45), then (41) holds, where is determined uniquely by (39).
Proof. It suffices to show that solves MF-FBDSDE (39).
Let ; we have where we have used Itô's formula and the equation satisfied by . Finally, let the mesh size go to zero; we have It is easy to check that