#### Abstract

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

#### 1. Introduction

Pardoux and Peng [1] in 1990 first introduced nonlinear classical backward stochastic differential equations (BSDEs). They proved the uniqueness and the existence of the solutions of nonlinear BSDEs under Lipschitz assumption. Since then the theory of BSDEs developed very fast and had found many applications, for example, in the stochastic control and partial differential equations. On the other hand, those stochastic Hamilton systems, derived from the stochastic maximum principle of stochastic optimal control problems, are forward-backward stochastic differential equations (FBSDEs).

The theory of fully coupled FBSDEs develops also very dynamically. There are many works on the existence and the uniqueness of solutions of fully coupled FBSDEs. Antonelli [2] first proved the existence and the uniqueness of solutions of fully coupled FBSDEs driven by Brownian motion on a small time interval with the fixed point theorem. There are also many other methods to study fully coupled FBSDEs on an arbitrarily given time interval, mainly three methods. One is “four-step scheme” approach (see Ma et al. [3]) which combines PDE methods and probability methods. The authors proved the existence and the uniqueness for fully coupled FBSDEs on an arbitrarily given time interval, but they required the diffusion coefficients to be nondegenerate and deterministic. Another one is purely probabilistic continuation method; refer to Hu and Peng [4], Pardoux and Tang [5], Peng and Wu [6], Yong [7], and so on. Another method is inspired by the numerical approaches for some linear FBSDEs (see Delarue and Menozzi [8] and Zhang [9]). There are also other methods; see Ma et al. [10]. For more details about fully coupled FBSDEs, the readers also refer to Ma and Yong [11] or Yong [7] and the references therein.

On the other hand, the theory of the modern optimal control has been developed widely since Pontryagin et al.’s work [12] about the maximum principle and Bellman’s work [13] on the dynamic programming approach. Later there have been a lot of works on the stochastic maximum principle; see, for example, Kushner [14, 15], Bensoussan [16], Haussmann [17], Peng [18], Wu [19], and so on. Wu [19] discussed the stochastic maximum principle for the fully coupled FBSDEs. Recently the methods of mean-field are used in various fields, such as in Finance, Chemistry, and Game Theory. The mean-field backward stochastic differential equations (mean-field BSDEs) were introduced by [20]; for more properties about mean-field BSDEs we refer to [21]. There are also many works on stochastic maximum principle for SDEs of mean-field type; see Andersson and Djehiche [22], Buckdahn et al. [23], Li [24], Bensoussan et al. [25], and so on. For more details we may refer to Yong [7].

In this paper, we consider the following fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs in short): where , , , take values in ; are mappings with appropriate dimensions which are -progressively measurable. The time duration is an arbitrarily fixed number. Our aim is first to find a triplet -adapted processes satisfying (1) and then study the stochastic maximum principle of mean-field FBSDEs with controls. For more works we refer to Qin [26].

In Section 2, we introduce the mean-field BSDEs. In Section 3, we prove the existence and the uniqueness of solution of mean-field FBSDE by the continuation method. In Section 4, we give the continuity of solutions of mean-field FBSDE with respect to the parameters and also give an example to show that our monotonicity conditions are necessary. In Section 5 we study the stochastic maximum principle for mean-field FBSDEs with controls and obtain the necessary condition of the stochastic maximum principle. In Section 6 we discuss mean-field backward stochastic linear quadratic optimal control problem as an example.

#### 2. Preliminaries

Let be a complete probability space with a standard -dimensional Brownian motion , and let be the natural filtration generated by and augmented by all P-null sets (i.e., , where is the set of all P-null subsets). is the fixed time horizon. .

Let be the (noncompleted) product of with itself. This product space is endowed with the filtration . A random variable originally defined on is extended canonically to . For any the variable is in -a.s., and its expectation is denoted by We notice that and

The generator of our mean-field BSDE is a mapping: which is -progressively measurable, for all , and satisfies the following assumptions.

We assume the following.

*(H2.1)*(i)is uniformly Lipschitz with respect to ;(ii); that is, is -valued-progressively measurable and.

Lemma 1. *Let (H2.1) hold, for any random variable ; the mean-field BSDE
**
has a unique solution ; that is, is -valued -adapted continuous process and ; is -valued -progressively measurable process and .*

For the proof, the readers may refer to [20].

*Remark 2. *From the above notions, the generator of the above mean-feld BSDE has to be understood as follows:

*Remark 3. *If we assume that(i) and are -progressively measurable continuous processes, for all and there exists some constant such that ,(ii) and are Lipschitz in ,then, for any random variable , the following mean-field SDE which is also the Mckean-Vlasov SDE has a unique adapted solution :
For more details, the reader may refer to, for example, [20] or [24].

#### 3. Mean-Field FBSDE: Existence and Uniqueness

We consider the following fully coupled mean-field forward-backward stochastic differential equations: where

*Remark 4. *In Li [24], the author studied the stochastic maximum principle in mean-field controls; the related feedback control system takes a special case of the mean-field FBSDE (8).

Given an full-rank matrix . We use the following notations: where . We use the standard inner product and Euclidean norm in .

*Definition 5. *A triple of processes is called an adapted solution of mean-field FBSDE (8), if , and satisfies mean-field FBSDE (8).

We assume the following.

*(H3.1)*(i)is uniformly Lipschitz with respect to ;(ii)for each is in ;(iii) is uniformly Lipchitz with respect to ;(iv)for each is in;(v)the coefficients are uniformly Lipschitz to .

We also need the following monotonicity assumptions.

*(H3.2)*(i);
(ii);
, , and are given nonnegative constants with (the equalities cannot be established at the same time), and ; , are the Lipschitz constants of with respect to , , respectively; and satisfies .

Or we need the following.

*(H3.3)*(i);
(ii);
, , and are given nonnegative constants with (the equalities cannot be established at the same time), and .

Then we have the following two main results in this section.

Theorem 6. *One assumes that (H3.1) and (H3.2) hold; then mean-field FBSDE (8) has a unique adapted solution .*

*Remark 7. *Similarly, if (H3.1) and (H3.3) hold, then mean-field FBSDE (8) has a unique adapted solution .

*Proof. *We first prove the uniqueness. Let and be two solutions of (8). We set , where , respectively. Applying Itô's formula to , we get
From (H3.2) the monotonicity assumptions of and , we get
When ,* ds dP*-a.e. In this case we have ,* P*-a.s., for all . Thus, , -a.s. Therefore, from Lemma 1 it follows that ,* P*-a.s. and ,* P*-a.s.a.e. When , thus ,* P*-a.s.a.e. From the uniqueness of solutions of McKean-Vlasov equations (refer to [20] or Remark 2), we get ,* P*-a.s., for all .

For the existence, we need to combine the above techniques and an a priori estimate to construct a contraction mapping. For this we first prove the following lemma.

For (the equalities cannot be established at the same time). We consider the following mean-field FBSDEs parameterized by : where and , and are given processes in with values in , and , respectively. . Obviously, when , the existence of (15) implies that of (8). From the existence and the uniqueness of Mckean-Vlasov equation and mean-field BSDE, (15) has a unique solution when . The following lemma is needed.

Lemma 8. *One assumes that (H3.1) and (H3.2) hold. If for an there exists a solution of (15), then there exists a positive constant such that for each there exists a solution of mean-field FBSDE (15) for .*

*Proof. *Since for every , there exists a (unique) solution of (15); for each and a triple there exists a unique triple
satisfying the following mean-field FBSDEs:
We want to prove that if is small enough, the mapping defined by is a contraction. Let and . We define .

Applying Itô’s formula to , it yields
where .

From (H3.1) and (H3.2), we know that if , then . Then, we have
On the other hand, from standard technique to the forward equation for , we get
From the above two estimates, we have
Here the constant depends on the Lipschitz constants, , and .

If , then . Then, we have
Then from the standard estimate of the mean-field BSDE part, we get
Here the constant depends on the Lipschitz constants, , and .

From the above two estimates and the standard estimate of , it follows that, for the sufficiently small ,
Here the constant depends only on the Lipschitz constants, , and .

From above all, we now choose . Obviously, for every fixed , the mapping is a contraction in the sense that
It means immediately that this mapping has a unique fixed point:
which is the solution of (15) for .

Now we can give the proof of the existence of the solution of mean-field FBSDE (8).

*Proof ( continued). *When , (15) has a unique solution. Then from Lemma 8, there exists a positive constant depending on Lipschitz constants, , and , such that, for every , (15) for has a unique solution. We can repeat this process times where . It means that, in particular, mean-field FBSDE (15) for has a unique solution; that is, (8) has a unique solution.

The proof is complete.

*Example 9. *We consider
The above FBSDE satisfes (H3.1) and (H3.2), form Theorem 6, we know it has a unique solution.

*Remark 10. *The proof of Remark 7 is similar. Notice that (15) should be changed into the following form:

*Remark 11. *When does not depend on , that is, is given, for the existence and the uniqueness of the solution of mean-field FBSDE (8), the monotonicity assumption (H3.2) can be weakened as
similarly, (H3.3) can be weakened as
where , , and are given nonnegative constants with , and , where the equalities cannot be established at the same time; are the Lipchitz constants of with respect to , , respectively; and satisfies .

Lemma 12. *When does not depend on , the mean-field FBSDE (8) also has a unique adapted solution, but the monotonicity (H3.2) should be weakened as*(i)*;
*(ii)*;
**similarly, (H3.3) can be weakened as*(i)*;
*(ii)*,
** where and are given nonnegative constants. Moreover, one has , , and .*

The proof of this lemma is similar to that of Theorem 6; we now only give the proof of the uniqueness.

*Proof. *Let and be two solutions of (8). We set , where , respectively. Applying Itô’s formula to , we get
From (H3.2) the monotonicity assumptions of and , we get
Applying Itô’s formula to , we get
where .

Then we get
Hence, we have
Thus, taking , we get
where .

Combining with (32), we have
When , , we have -a.e. In this case we have ,* P*-a.s., for all . Thus, , -a.s. Therefore, from Lemma 1 it follows that ,* P*-a.s. and ,* P*-a.s.

#### 4. Continuity Property on the Parameters

In this section we will discuss the continuity of the solution of (8) depending on parameters. We consider the following mean-field FBSDEs with coefficients : where , , and , satisfy (H3.1) and (H3.2) for each . Then, from Theorem 6 we know that mean-field FBSDE (38) has a unique solution for each .

Let us give some more assumptions.

*(H4.1)*(i)The coefficients, are uniformly Lipschitz to ;(ii)the mappings, are continuous, respectively.

Then we have the following continuity property.

Theorem 13. *Let the coefficients , satisfy (H3.1), (H3.2), and (H4.1), and the associated solution of mean-field FBSDE (38) is denoted by . Then, the mappings
**
are continuous.*

*Proof. *For simplicity of notations, we only prove the continuity of the solutions of mean-field FBSDE (38) at . We want to prove that converges to in as tends to 0. We set and ; then from (38) we know that
From assumptions (H3.1), (H3.2), and (H4.1) and standard estimates of and , we get
where depends on the Lipchitz constants and , where
Applying Itô’s formula to , it yields
where
Then, we have
From (H3.2) we know if , then . Then, from (46) we have
With the help of (42) and (47) we can take sufficiently small such that
where the constant only depends on .

Similarly, if , then . Then, from (46) we have
With the help of (41) and (49) we can take sufficiently small such that
where the constant only depends on .

Hence, we have that converges to in as tends to 0.

Now we will give an example to explain that (H3.2) (or (H3.3)) is necessary; that is, if the coefficients do not satisfy (H3.2) (or (H3.3)), then (8) may not have a solution. We take here. We consider It is easy to check that this equation does not satisfy (H3.2) (or (H3.3)); we point out that, there is also no adapted solution. In fact, if is the solution of mean-field FBSDE (51), then is the solution of the following ordinary differential equation: