Maximum Principle for Mean-Field FBSDEs with Mixed Initial-Terminal Conditions

Li, Ruijing

doi:https://doi.org/10.1155/2022/6019460

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 6019460 | https://doi.org/10.1155/2022/6019460

Maximum Principle for Mean-Field FBSDEs with Mixed Initial-Terminal Conditions

Ruijing Li¹

Academic Editor: Binchang Wang

Received09 Feb 2022

Accepted18 Apr 2022

Published10 May 2022

Abstract

The present paper concerns with optimal control problems allowing for time inconsistent utility functions of mean-field FBSDEs with mixed initial-terminal conditions. Moreover, the control variable enters the diffusion coefficient, and the control domain is with nonconvexity. Via extended Ekeland’s variational principle as well as the reduction method, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.

1. Introduction

It is well known that the key tools in solving optimal control problems are based upon Pontryagin’s maximum principle and the Bellman dynamic programming principle. Recently, the mean-field approaches have been widely applied in various fields, such as in Finance, Statistical Mechanics, and Game theory. The systems these fields involve are mainly of mean-field type, where the performance functionals, drifts, and diffusion coefficients depend not only on the state but also on the probability distribution of the state. In this case, the mean-variance utility functions are of time inconsistence, for which the Bellman dynamic programming principle does not hold. Therefore, problems involving such utilities cannot be approached by the classical Hamiltonian–Jacobi–Bellman equation. In this paper, we study a mean-variance setting of the Principle-Agent problem by means of Pontryagin’s maximum principle.

The pioneering paper in which nonlinear mean-field backward stochastic differential equations (BSDEs) first appeared is in [1]. Taking advantage of a BSDE approach, the paper [2] considered the mean-field BSDE and gave a probabilistic interpretation to related nonlocal partial differential equations. Since then, the theory of mean-field forward-backward stochastic differential equations (FBSDEs) has been intensively investigated, and rich results have been obtained for the system of this kind of McKean–Vlasov type (see [3]) in the literature. For instance, by using the penalization method to construct approximation solution, the authors in [4] proved the existence of solutions to first-order mean-field games arising in optimal switching. In [5], by translating the splitting method to BSDEs, the author proved the existence and the uniqueness of the solution of the split equations; moreover, under appropriate regularity assumptions on the coefficients, the value function turned out to be the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation of mean-field type. On the other hand, with the development of mean-field FBSDEs, research on stochastic control problems in mean-field context got rapid growth. For instance, by utilizing a spike variation of the optimal control, the authors in [6] established Peng’s-type stochastic maximum principle. When the controlled system contains the Markov jump parameters, the authors in [7] established the corresponding maximum principle for mean-field stochastic linear-quadratic optimal control problems. Furthermore, with the help of maximum principle, the authors in [8] presented necessary as well as sufficient solvability condition of the finite horizon mean-field optimal control problem in an explicit expression form, and the author in [9] established necessary and sufficient optimality conditions for the mean-field jump-diffusion system with moving-average as well as pointwise delays. Further, by using a backward separation method with a dominated growth rate as well as an approximate technique, the authors in [10] presented a maximum principle for optimality of mean-field stochastic differential equations, where the state is partially observed via a noisy process. For some related topics, we refer the readers to [4, 11–13], [14], [7, 15, 16], [17], and the references therein.

However, to the best of our knowledge, when the control domain is not assumed convexity, for the stochastic optimal control problems with mixed initial-terminal constraints, their corresponding stochastic maximum principle is all established on the basis of requirements of lower semicontinuity of the penalty functional. So, how to reduce this requirement is a new problem. This paper is trying to remedy this issue and establish stochastic maximum principle for mean-field FBSDEs under general control domains. With the help of extended Ekeland’s variational principle and the reduction method, a general stochastic maximum principle will be acquired. It is necessary to point out that, due to the application of extended Ekeland’s variational principle, the reduction method we adopt is different from those given in [18]; meantime, our results extend those of [19] essentially to the framework of mean-field theory.

The rest of this paper is organized as follows. Preliminaries are in Section 2. In Section 3, we give some prior estimates and establish the general stochastic maximum principle in the framework of mean-field theory. In Section 4, a linear-quadratic stochastic control problem is worked out to illustrate the theoretical results. Finally, some concluding remarks are given in Section 5.

2. Preliminaries

Let be a filtered probability space satisfying the usual condition, on which a one-dimensional Brownian motion is defined, and be the natural filtration generated by and augmented by all -null sets, i.e.,where is the set of all -null subsets. We denote by the set of -adapted and continuous stochastic processes such that , by the set of -adapted stochastic processes such that , and .

Let be the (noncompleted) product of with itself, and this product space is endowed with the filtration . Then, any random variable originally defined on can be extended canonically to : , . For any , the variable : belongs to , -., and its expectation is denoted by

Notice that , and .

Consider a fully coupled mean-field stochastic differential equation with mixed initial-terminal conditions as follows:where ; ; is a nonempty closed and nonconvex subset of .

Remark 1. The drift terms and and the diffusion term of (3) have to be interpreted as follows:The cost functional associated with (3) is of the form as follows:where ; . The admissible control set is defined asThen, the optimal control problem we are concerned with is as follows.

Problem 1. Find a such thatIf attains the infimum, then it is called optimal; the solution corresponding to is called an optimal trajectory. In what follows, some notations and assumptions shall be assumed to ensure the well-posedness of stochastic system (3). We denote the scalar product by and the norm by of an Euclidean space. For , and define , . The functions , , , , , , and are twice continuously differentiable with respect to . Moreover, all their derivatives with respect to up to second-order are continuous in and bounded. The functions , , , , , , and grow linearly with respect to and are continuous in , respectively. For any , , there exists a constant such that (monotonic conditions). , , , , where , and , are given nonnegative constants.

Remark 2. Under the assumptions or , via Theorem 1 in [20], the mean-field system (3) admits a unique adapted solution .
In fact, due to the mixed initial-terminal conditions in the state equation, even if the well-posedness of the state equation is ensured via the Lyapunov operator introduced in [21], the well-posedness of the first-order adjoint equation seems to be not guaranteed. On the other hand, since both and appear in the coefficients of the backward equation (3), the regularity of process (as a part of the state process) is not enough to obtain a second-order adjoint equation. Thus, the classical method cannot be directly used. To overcome this difficulty, we introduce a reduction method inspired by the study of optimality variational principle for controlled FBSDEs [18]. First, a variation of problem is formulated as follows.

Problem 2. Minimizesubject toOver with the mixed initial-terminal state constraints,

Remark 3. Via Theorem 2 in [22], under the assumptions , the mean-field forward stochastic system (10) admits a unique solution .
It is remarkable that problem is embedded into problem . Hence, if is the optimal control of , then is optimal for . In the following section, the classical second-order variational technique will be adopted to solve problem .
Before presenting the main results, we introduce the extended Ekeland’s variational principle, which plays an important role in obtaining the stochastic maximum principle.

Theorem 1. (extended Ekeland’s variational principle [23]). Let be a nonempty closed subset of a complete metric space and be a bifunction. Assume that and the following assumptions are satisfied:(1) For all , is closed(2), for all (3), for all If for some , then there exists such that

3. Stochastic Maximum Principle

In this section, we will try to solve problem . Firstly, suppose is an optimal control of , with the corresponding optimal state process . Without loss of generality, we assume that . For any and , the penalty functional is defined as

Endow the set and with the metric ,where denotes the Lebesgue measure. Then, it is easy to verify as in [24] that is a complete metric space with the metric :

Clearly, (13) subject to (10) is a forward stochastic control problem without the state constraints. However, we have to notice that the unboundedness of cannot assure lower semicontinuity of in . So, Ekeland’s variational principle cannot be directly used. Fortunately, extended Ekeland’s variational principle can reduce the requirement of lower semicontinuity of . Hence, Theorem 1 can be adopted to discuss this issue. First, we consider the case that , , and take values in , , and , and is convex. Moreover, and are all closed.

Set with and . Then, satisfies the conditions of Theorem 1, the demonstration process is similar as the appendix in [25], and thus the detail is omitted. Hence, there exists such that

Note that, is the optimal control variables for problem ; they satisfy the optimal state constraints at time . Thus, from the definition of penalty functional (13), we can derive . Further, we can get

Then, (17) means that the control process is a global minimum point of the following penalized cost functional:

Since and are dynamic, spike technique for and is employed. For sufficiently small , define

For some with , where also denotes the Lebesgue measure, . For and , since they are independent of the time variable and range in a convex control domain, we adopt the convex perturbation for and . Let be control variables such that and . Then, for any , and are also control variables with values in .

For the sake of convenience, we denote

These notations also work for their corresponding derivatives.

Suppose and are solutions of (10) corresponding to and , respectively. We introduce the first-order and second-order variational equations.where denotes the indicator function of a set . It is easy to verify that (21) and (22) admit unique solutions, and these solutions satisfy the following prior estimates.

Lemma 1. Let or hold, thenwhere such that .

Proof 1. The detail refers to Lemma 1 in [6].

Theorem 2. Suppose or hold. is an optimal control for with the corresponding optimal trajectory . Then, there exist three nonnegative parameters , , and with and an adapted solution of the following FBSDE:such thatandwhere is the Hamiltonian function defined byAnd is the unique adapted solution of the following BSDE:

Proof 2. This state constrained stochastic maximum principle will be proved into two steps.

Step 1. Let , and take values in , , and , respectively, and be convex. Moreover, and are all closed.
From (17) and the definition of , we haveBy using Taylor expansion with addition of Lemma 1, we deducewhere stands for certain vectors that tend to 0 as and stands for inner product of vectors. Here, , , andThen, and . Thus, there exists a subsequence still denoted by convergent, i.e.,We claim that . The detailed illustration of this point refers to [19]. Here, is called the Lagrange multiplier of the corresponding optimal 4-tuple . While, by Taylor expansion,whereSimilarly,where .
Consequently, from , the variational inequality is derived asTo get rid of in the above inequality, we introduce the first-order adjoint equations asApplying Itô’s formula to fulfillsCombining (36) and (38), again with the help of Lemma 1, the variational inequality (36) can be changed aswith being the Hessian ofWe see that in (39), there left some second-order terms in . Hence, we want to further get rid of them. Firstly, the equations satisfied by are presented by appealing to Itô’s formula.whereSecondly, we introduce the second-order adjoint equations corresponding to , respectively, to remove in (39). Now, let be the adapted solution of the following BSDE:Then, by Lemma 1 and applying Itô’s formula to yieldInserting (44) into (39), we can derivewhereBy the Frechet differentiability, for , we haveHence,Note that ; there exists a subsequence of still denoted by such that . Consequently, by the continuous dependence of the solution of mean-field BSDEs on parameters, we have the following conclusions, as , where and are solutions of first-order and second-order adjoint equations corresponding to , respectively. Hence, let and send , , we see thatwhich impliesAlso, dividing in (45) and sending , , we gettwhich implies that the necessary condition (25) holds. Finally, by taking in (45) and sending , then using a standard argument, the variational inequality (26) follows.
Finally, letThen, it follows from (37) and (50) thatBy use of (52), we may rewrite the first-order adjoint (37), further giving the one-dimensional first-order adjoint (24).

Step 2. The general case of control domains.
First, we setIt is easy to see that is convex andThen, by using the similar methods in [25], the desired conclusion (26) can be drawn.
Since is arbitrary, we draw the desired conclusion and summarize it as follows.

Theorem 3 (general maximum principle). Suppose or hold. is an optimal tuple for problem with the corresponding optimal trajectory . Then, there exist three nonnegative parameters , , and with , such that for any , , , and , the necessary conditions (25) and (26) hold a.e. a.s., where is the Hamiltonian function and are solutions of (24) and (28) corresponding to , respectively.

Remark 4. When the coefficients , , and do not depend on , system (3) reduces to the standard one, then Theorem 3 reduces to the necessary conditions for classical fully coupled FBSDEs, and the corresponding results were obtained by [19]. So our result is nontrivially more general of [19].

Remark 5. Compared with reference [19], the extended Ekeland’s variational principle is firstly introduced to reduce the requirement of lower semicontinuity of . Secondly, a convergence technique is used to establish the stochastic maximum principle in general control domains. These methods are all different from the ones in [19].

Remark 6. For tackling the mixed initial-terminal conditions, a reduction method is adopted to transform the original mixed initial-terminal conditions as mixed initial-terminal state constraints. This treatment not only ensures the well-posedness of the first-order adjoint equations but also simplifies the derivation of the second-order adjoint equations.

4. Application

Consider the optimal control problem.

Problem 3. Minimize over , subject towhere , , , and , , , are real constants of . Moreover, , , , and . In practice, the range of control variable is limited for some reasons such as (see [18]). Assume thatis negative definite matrix, andis positive definite matrix. Then, for any , there exists a unique state process . Now, suppose is optimal for problem , then the adjoint equations are read as follows:Clearly, (59) admits unique solution by assumptions . And the BSDE for becomeswhere and . According to the existence and uniqueness of solutions,Hence, inequality (25) holds naturally. Then, let in the maximum condition (26); a candidate optimal control should satisfyThen, we havewith .

Remark 7. From the above proof process, we note that when the coefficients of system (56) only depend on the expected values of the solution states, the second-order adjoint equation is mean-field independent and solvable.

Proposition 1. The control defined by (63) is optimal, together with the corresponding trajectory which is an optimal solution for problem .

Proof 3. Suppose is the trajectory of system (56) controlled by . By the convexity of a function, we haveandApplying Itô’s formula to , one hasTaking notice of (64)–(66), From (63), we know thatholds for all . This means that , , which yields the optimality of .

5. Conclusion

This paper discussed optimal control problems of mean-field FBSDEs with mixed initial-terminal conditions. Firstly, we initially introduce extended Ekeland’s variational principle to reduce requirements of lower semicontinuity of the penalty functional. Secondly, the reduction method is adopted to guarantee the well-posedness of the first-order and second-order adjoint equations also with mixed initial-terminal conditions. Via spike variational technique, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.

Data Availability

Data sharing is not applicable to this article as no datasets were generated during the current study.

Conflicts of Interest

The author declares that she has no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was partially supported by National Natural Science Foundation of China (Grant no. 11901112).

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Copyright © 2022 Ruijing Li. 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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