Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

Min, Hui; Peng, Ying; Qin, Yongli

doi:https://doi.org/10.1155/2014/839467

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 839467 | https://doi.org/10.1155/2014/839467

Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

Hui Min,¹Ying Peng,²and Yongli Qin^1,3

Academic Editor: Michel Chipot

Received21 Dec 2013

Accepted04 Feb 2014

Published13 Apr 2014

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: But we know this ODE has no solution; therefore, there is no adapted solution of (51).

5. Maximum Principle for the Controlled Mean-Field FBSDEs

We consider the following controlled mean-field fully coupled forward-backward SDEs: where , takes value in . Let be a nonempty convex subset of An element of is called an admissible control. We define the following cost functional: where The optimal control problem is to minimize the cost functional over all admissible controls. An admissible control is called an optimal control if the cost functional attains the minimum at . Equation (53) is called the state equation; the solution corresponding to is called the optimal trajectory.

We assume the following.

(H5.1)(i)and are continuously differentiable to;(ii)the derivatives of are bounded;(iii)the derivatives ofare bounded by;(iv)the derivatives ofandare bounded byandrespectively;(v)for any given admissible controlthe coefficients satisfyand.

Let be an optimal control and let be the corresponding optimal trajectory. Let be such that . Since is convex, then for any is also in .

We introduce the following linear mean-field FBSDE: where .

Remark 14. When , respectively, is the partial derivative of with respect to ; is the partial derivative of with respect to , similar to .
From (H5.1), it is easy to verify that (57) satisfies (H3.1) and (H3.2); then there exists a unique solution of mean-field FBSDE (57). Equation (57) is called the variational equation.
We denote by the trajectory corresponding to . Then we have the following convergence result.

Lemma 15. One assumes (H5.1) holds. Then, , , and , in .

Proof. Let . Then , , and ,. From Theorem 13, it is easy to know that converges to in as tends to . Now, we define
Then, The above Equation(60) can be rewritten as the following: where where , respectively, and From (H5.1) and the fact that converges to 0 in as tends to 0, we know that where , , . has similar results.
As we know, (57) has a unique solution . Therefore, the solution converges to in as tends to .

Because is an optimal control, then From (65) and Lemma 15, we have the following.

Lemma 16. One supposes that (H5.1) holds. Then, the following variational inequality holds: where .

Proof. Let in (65); from Lemma 15 and (H5.1), it is obvious that The proof is complete.

Now we introduce the following adjoint mean-field FBSDE to (57): where , , and . From Theorem 6, we know that there exists a unique triple satisfying (68).

We define the Hamiltonian function as follows: Then we have the following maximum principle.

Theorem 17. Let be an optimal control and let be the corresponding trajectory. Then, one has where , is the solution of the adjoint (68).

Proof. Applying Itô’s formula to , from (57) and (68) and (H3.1), (H3.2), (H3.3), and (H5.1), with the help of (66) and (69), for such that , we get Therefore, we have

6. Application to the Mean-Field LQ Problems

In this section, we consider a linear-quadratic control problem as an example. For simplicity, we only consider one-dimensional case; that is, . The state equation can be written as follows: where the constants , are positive and . The cost functional is where are bounded and nonnegative and is bounded and positive. Then, the adjoint equation is the following mean-field FBSDE: Let

Then, from the stochastic maximum principle (Theorem 17), we have that is, However, the maximum principle gives only the necessary condition for optimal control.

Now we prove that is the optimal control. For all , let be the corresponding trajectory. Then We apply Itô’s formula to , where is the solution of adjoint equation with the state process ; notice that when is a constant, then and we get Therefore, from the definition of , for any . It means that is an optimal control.

Remark 18. Under our assumption, the existence and the uniqueness of the solution of (73) and (75) can be obtained by combining the method of Theorem 3.1 and Theorem 2.1 in [6]. We omit the proof.

Conflict of Interests

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

Acknowledgments

The work has been supported by the NSF of China (nos. 11071144, 11171187, and 11222110), Shandong Province (nos. BS2011SF010 and JQ201202), Program for New Century Excellent Talents in University (no. NCET-12-0331), and the 111 Project (no. B12023).

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Copyright © 2014 Hui Min et al. 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.

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