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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 [

Peng and Shi [

McKean-Vlasov stochastic differential equation (SDE) of the form

Mean-field backward doubly stochastic differential equations (MF-BDSDEs) of the form

In this paper, we would like to introduce mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) of the form

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 [

The paper is organized as follows. In Section

Let

Let

We would like to introduce some spaces of functions required in the sequel:

We will give notations as follows:

Let

Consider the following MF-FBDSDEs:

A quadruple of

One assumes the following.

(H1) For each

(H2)

The following monotonic conditions, introduced in [

(H3)

In order to prove the existence and uniqueness result for (

When

Under assumptions (H1)–(H3), there exists a positive constant

Let

Applying Itô’s formula to

On the other hand, for the difference of the solutions

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 (

Under assumptions (H1)–(H3), (

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

(H3)′

By similar arguments to Theorem

Under assumptions (H1), (H2), and (H3)^{’}, MF-FBDSDE (

The connection of BDSDEs and systems of second-order quasilinear SPDEs was observed by Pardoux and Peng [

For each

To simplify the notation, for

If there exists

Assume that

It suffices to show that

Let

In the case when

Equation (

Equation (

By virtue of a connection between them and fully coupled FBDSDE of mean-field type, Theorem

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

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).