Existence and uniqueness result of the solutions to mean-field backward doubly stochastic differential equations (BDSDEs in short) with locally monotone coefficients as well as the comparison theorem for these equations are established. As a preliminary step, the existence and uniqueness result for the solutions of mean-field BDSDEs with globally monotone coefficients is also established. Furthermore, we give the probabilistic representation of the solutions for a class of stochastic partial differential equations by virtue of mean-field BDSDEs, which can be viewed as the stochastic Feynman-Kac formula for SPDEs of mean-field type.
Keywords:mean-field; backward doubly stochastic differential equations; locally monotone coefficients; comparison theorem; stochastic partial differential equations
In this paper, we study a new kind of stochastic partial differential equations (SPDEs):
Here, the function is the unknown function, and is an l-dimensional Brownian motion process defined on a given complete probability space . , a stochastic process starting at when , is the solution of one class of stochastic differential equations (SDEs), and E denotes expectation with respect to the probability P. In this paper, we call this kind of equations (1.1) McKean-Vlasov SPDEs, because they are analogous to McKean-Vlasov PDEs except the stochastic term .
McKean-Vlasov PDEs involving models of large stochastic particle systems with mean-field interaction have been studied by stochastic methods in recent years (see [1-4] and the references therein). Mean-field approaches have applications in many areas such as statistical mechanics and physics, quantum mechanics and quantum chemistry. Recently, Lasry and Lions introduced mean-field approaches for high-dimensional systems of evolution equations corresponding to a large number of ‘agents’ or ‘particles’. They extended the field of such mean-field approaches to problems in economics, finance and game theory (see  and the references therein).
As is well known, to give a probabilistic representation (Feynman-Kac formula) of quasilinear parabolic SPDEs, Pardoux and Peng  introduced a new class of backward stochastic differential equations (BSDEs) called backward doubly stochastic differential equations which have two different types of stochastic integrals: a standard (forward) stochastic integral and a backward stochastic integral . They proved the existence and uniqueness for solutions of BDSDEs under uniformly Lipschitz coefficients. When the coefficients are smooth enough, they also established the connection between BDSDEs and a certain kind of quasilinear SPDEs. BDSDEs have a practical background in finance. The extra noise B can be regarded as some extra inside information in a derivative security market. Since 1990s, BDSDEs have drawn more attention from many authors (cf.[7-13] and the references therein). Shi, Gu and Liu gave the comparison theorem of BDSDEs and investigated the existence of solutions for BDSDEs with continuous coefficients in . To relax the Lipschitz conditions, Wu and Zhang studied two kinds of BDSDEs under globally (respectively, locally) monotone assumptions and obtained the uniqueness and existence results of the solutions (see ).
Mean-field BSDEs are deduced by Buckdahn, Djehiche, Li and Peng  when they studied a special mean-field problem with a purely stochastic method. Later, Buckdahn, Li and Peng  investigated the properties of these equations in a Markovian framework, obtained the uniqueness of the solutions of mean-field BSDEs as well as the comparison theorem and also gave the viscosity solutions of a class of McKean-Vlasov PDEs in terms of mean-field BSDEs.
In this paper, we study a new type of BDSDEs, that is, the so called mean-field BDSDEs, under the globally (respectively locally) monotone coefficients. We obtain the existence and uniqueness result of the solution by virtue of the technique proposed by Wu and Zhang  and the contraction mapping theorem under certain conditions. Also, the comparison principle for mean-field BDSDEs is discussed when the coefficients satisfy some stricter assumptions. A comparison theorem is a useful result in the theory of BSDEs. For instance, it can be used to study viscosity solutions of PDEs. Here, we point out that it is more delicate to prove the comparison theorem for mean-field BDSDEs because of the mean-field term.
In Eq. (1.2), the integral is a forward Itô integral, and the integral denotes a backward Itô integral. and are two mutually independent standard Brownian motion processes with values respectively in and in . This conclusion gives a probabilistic representation of McKean-Vlasov SPDEs (1.1), which can be regarded as a stochastic Feynman-Kac formula for Mckean-Vlasov SPDEs.
Our paper is organized as follows. In Section 2, we present the existence and uniqueness results about mean-field BDSDEs with globally monotone coefficients. We investigate the properties of mean-field BDSDEs with locally monotone assumptions in Section 3. We first prove the existence and uniqueness of the solutions of mean-field BDSDEs and then derive the comparison theorem when the mean-field BDSDEs are one-dimensional. In Section 4, we introduce the decoupled mean-field forward-backward doubly stochastic differential equation and study the regularity of its solution with respect to x, which is the initial condition of the McKean-Vlasov SDE. Finally, Section 5 is devoted to the formulation of McKean-Vlasov SPDEs and provides the relationship between the solutions of SPDEs and those of mean-field BDSDEs.
2 Mean-field BDSDEs with globally monotone coefficients
In this section, we study mean-field BDSDEs with globally monotone coefficients, which is helpful for the case of locally monotone coefficients. To this end, we firstly introduce some notations and recall some results on mean-field BSDEs obtained by Buckdahn, Li and Peng .
Let and be two mutually independent standard Brownian motion processes, with values respectively in and , defined over some complete probability space , where T is a fixed positive number throughout this paper. Moreover, let denote the class of P-null sets of ℱ. For each , we define
We will also use the following spaces:
Let be the (non-completed) product of with itself, and we define on this product space. A random variable originally defined on Ω is extended canonically to , . For any , the variable belongs to , -a.s., whose expectation is denoted by
We now consider the following mean-field BDSDEs with the form:
Remark 2.1 Due to our notation, the coefficients of (2.1) are interpreted as follows:
then it must satisfy conditions (A4) and (A5).
The main result of this section is the following theorem.
has a unique solution. In order to get this conclusion, we define
Then (2.2) can be rewritten as
and f fulfills
According to Theorem 2.2 in , BDSDE (2.2) has a unique solution.
Then we have
Consequently, I is a strict contraction on equipped with the norm for . With the contraction mapping theorem, there admits a unique fixed point such that . On the other hand, from Step 1, we know that if , then , which is the unique solution of Eq. (2.1). □
The proof of the above corollary is similar to that of Theorem 2.1 and is therefore omitted.
3 Mean-field BDSDEs with locally monotone coefficients
In this section, we investigate mean-field BDSDEs with locally monotone coefficients. The results can be regarded as an extension of the results in  to the mean-field type.
We need the following lemma, which plays an important role in the proof of the main result.
where is a sequence of smooth functions such that , for , and for . Similarly, we define the sequences , , . It should be pointed out that , , and are continuously differentiable with bounded derivatives for each m. The conclusion of this lemma can be easily obtained by arguments similar to those of Lemma 3.3 in . □
We now present the main result of this section.
Theorem 3.1Let (A1), (A2), (A3′)-(A5′) hold. Assume, moreover,
Proof We now construct an approximate sequence. Let be associated to f by Lemma 3.1. Then for each m, is globally monotone in y and globally Lipschitz in z. By Theorem 2.1, the following mean-field BDSDE
Then it follows from Gronwall’s inequality and the B-D-G inequality that
We next estimate I, II and III.
For the first term I, based on Hölder’s inequality and Chebyshev’s inequality, we have
For the last term, we have
Applying Gronwall’s inequality and the B-D-G inequality to the above inequality yields
then we have
Therefore, we come to the conclusion of this theorem. □
Theorem 3.2 (Comparison theorem)
Assume mean-field BDSDEs (3.9) and (3.10) satisfy the conditions of Theorem 3.1. Letandbe the solutions of mean-field BDSDEs (3.9) and (3.10), respectively. Moreover, for the two generators ofand, we suppose:
Then we have
With the assumption (A1), we obtain
Combining (3.12), (3.13) with (3.11) yields
By Gronwall’s inequality, it follows that
4 Decoupled mean-field forward-backward doubly SDEs
In this section, we study the decoupled mean-field forward-backward doubly stochastic differential equations. First, we recall some results of Buckdahn, Li and Peng  on McKean-Vlasov SDEs. Given continuous functions and which are supposed to satisfy the following conditions:
From the result about Eq. (5.1) in , we know that under Assumption 4.1, SDE (4.1) has a unique strong solution, and we can obtain that has a continuous version with the following well-known standard estimates.
Next, we investigate the solution of the following BDSDE:
and . Then due to Theorem 2.2 in , we obtain that there exists a unique solution to Eq. (4.3).
For BDSDE (4.3), we give the following proposition.
Proof Combining classical BDSDE estimates (see the proof of Theorem 2.1 in Pardoux and Peng ) with the techniques presented in Theorem 3.1, we can get the proof easily. □
5 Mean-field BDSDEs and McKean-Vlasov SPDEs
In fact, Eq. (5.1) is a new kind of nonlocal SPDE because of the mean-field term. Here, the functions b, σ, f, g and Φ are supposed to satisfy Assumption 4.1 and Assumption 4.2 respectively, and is the solution of the mean-field SDE (4.1) with .
Now, we give the main theorem of this section.
According to our notations introduced in Section 2, we know that
Then we have
Remark 5.1 Formula (5.2) generalizes the stochastic Feynman-Kac formula for SPDEs of the mean-field type.
The author declares that he has no competing interests.
The author would like to thank the editor and anonymous referees for their constructive and insightful comments on improving the quality of this revision, and to thank professor Zhen Wu for many helpful suggestions.
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