Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs

Abstract

We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs) in Hilbert spaces under a weaker condition than the Lipschitz one. As an intermediate step, the existence and uniqueness result for the mild solutions of mean-field BSEEs under Lipschitz condition is also established. And then a maximum principle for optimal control problems governed by backward stochastic partial differential equations (BSPDEs) of mean-field type is presented. In this control system, the control domain need not to be convex and the coefficients, both in the state equation and in the cost functional, depend on the law of the BSPDE as well as the state and the control. Finally, a linear-quadratic optimal control problem is given to explain our theoretical results.

1. Introduction

Backward stochastic evolution equations (BSEEs) in their general nonlinear form were introduced by Hu and Peng [1] in 1991. By the stochastic Fubini theorem and an extended martingale representation theorem, Hu and Peng [1] obtained the existence and uniqueness result of a so-called “mild solution” under Lipschitz coefficients for semilinear BSEEs. Since then, BSEEs have been studied by a lot of authors and have found various applications, namely, in the theory of infinite dimensional optimal control and the controllability for stochastic partial differential equations (see e.g., [1–4] and the papers cited therein). To relax the Lipschitz condition of the coefficients, Mahmudov and Mckibben [2] studied BSEEs under a weaker condition than the Lipschitz one in Hilbert spaces. Their approach extended the method proposed by Mao [5], in which the author investigated BSDEs under a weaker condition which contains Lipschitz condition as a special case. Our present work also investigates backward stochastic evolution equations, but with one main difference to the setting chosen by the papers mentioned above: the coefficients of the BSEEs are allowed to depend on the law of the BSEEs.

Recently, mean-field approaches, which can be used to describe particle systems at the mesoscopic level, have attracted more and more researchers’ attention because of their great importance in applications. For example, mean-field approach can be used in statistical mechanics and physics, quantum mechanics and quantum chemistry, economics, finance, game theory, and optimal control theory (refer to [6–8] and the references therein). Mean-field BSDEs were deduced by Buckdahn et al. [9] when they investigated a special mean-field problem in a purely stochastic approach. Buckdahn et al. [7] studied the well posedness of mean-field BSDEs and gave a probabilistic interpretation to semilinear McKean-Vlasov partial differential equations. To give a probabilistic representation of the solutions for a class of Mckean-Vlasov stochastic partial differential equations, Xu [10] investigated the well-posedness of mean-field backward doubly stochastic differential equations with locally monotone coefficients.

In this paper, we investigate a new type of backward stochastic evolution equations in Hilbert spaces which we call mean-field BSEEs. Mean-field implies that the coefficient of the BSEE depends on the law of the BSEE. Specifically, the BSEE we consider is defined as in a Hilbert space , where denotes a given measurable mapping, is a fixed positive real number, is a cylindrical Wiener process, and represents the generator of a strongly continuous semigroup in with . Precise interpretation of is given in the following sections. Based on the contraction mapping, we firstly prove that (1) admits a unique mild solution if the function is Lipschitz continuous. Secondly, under non-Lipschitz assumptions, we obtain the existence and uniqueness of the mild solution for mean-field BSEE by constructing a special Cauchy sequence. The Lipschitz condition is a special case of this non-Lipschitz condition (see Mao [5]). In addition, we investigate the well-posedness of mean-field stochastic evolution equations.

We also study optimal control problems of stochastic systems governed by mean-field BSPDEs in Hilbert spaces. Our objective is to formulate a stochastic maximum principle (SMP) for the optimal control problem with an initial state constraint. There is a vast literature on the theory of SMP. Among these papers, Andersson and Djehiche [8] studied the optimal control problem for mean-field stochastic system when the control domain is convex. They obtained the maximum principle by a convex variational method. By a spike variational technique, Buckdahn et al. [11] obtained a general maximum principle for a special mean-field stochastic differential equation (SDE) where the action space is not convex. Later, Li [12] investigated the maximum principle for more general SDEs of mean-field type with a convex control domain. Wang et al. [13] were concerned with a partially observed optimal control problem of mean-field type. By using Girsanov’s theorem and convex variation, they derived the corresponding maximum principle and gave an illustrative example to demonstrate the application of the obtained SMP. Hafayed studied the mean-field SMP for singular stochastic control in [14] and mean-field SMP for FBSDEs with Poisson jump processes in [15].

For the case of stochastic control systems in infinite dimensions, on the assumption that the control domain is not necessarily convex while the diffusion coefficient does not contain the control variable, Hu and Peng [16] used spike variation approach and Ekeland’s variational principle to establish the maximum principle for semilinear stochastic evolution control systems with a final state constraint. Mahmudov and Mckibben [2] obtained an SMP for stochastic control systems governed by BSEEs in Hilbert spaces. Recently, Fuhrman et al. [17] deduced the maximum principle for optimal control of stochastic PDEs when the control domain is not necessarily convex.

We establish necessary optimality conditions for the control problem in the form of a maximum principle on the assumption that the control domain is not necessarily convex. Due to the initial state constraint, we first need to apply Ekeland’s variational principle to convert the given control problem into a free initial state optimal control problem. Then spike variation approach is used to deduce the SMP in the mean-field framework. In our control system, not only the state processes which are the unique mild solution of the given BSPDE, but also the cost functional are of mean-field type. In other words, they depend on the law of the BSPDE as well as the state and the control. For this new controlled system, the adjoint equation will turn out to be a mean-field stochastic evolution equation.

The plan of this paper is organized as follows. In Section 2, we introduce some notations which are needed in what follows. In Section 3, the well-posedness of mean-field BSEE (1) is studied; we first prove the existence and uniqueness of a mild solution under the Lipschitz condition and investigate the regular dependence of the solution on . And then, under the assumption that the coefficient is non-Lipschitz continuous, a new result on the existence and uniqueness of the mild solution to (1) in Hilbert space is established, which generalizes the result for the Lipschitz case. Section 4 is devoted to the regularity of mean-field stochastic evolution equations. In Section 5, we derive the stochastic maximum principle for the BSPDE systems of mean-field type with an initial state constraint, and at the last part of Section 5, an LQ example is given to show the application of our maximum principle. An explicit optimal control is obtained in this example.

2. Preliminaries

The norm of an element in a Banach space is denoted by or simply , if no confusion is possible. , , and are three real and separable Hilbert spaces. Scalar product is denoted by , with a subscript to specify the space, if necessary. is the space of Hilbert-Schmidt operators from to , endowed with the Hilbert-Schmidt norm.

Let be a complete probability space. A cylindrical Wiener process in a Hilbert space is a family of linear mappings such that(i)for every , is a real (continuous) Wiener process;(ii)for every , and , , .

By , , we denote the natural filtration of , augmented with the family of -null sets of : The filtration satisfies the usual conditions. All the concepts of measurability for stochastic processes (e.g., adapted, etc.) refer to this filtration.

Next we define several classes of stochastic processes with values in a Hilbert space .(I) denotes the set of (classes of a.e. equal) measurable random processes which satisfy(i),(ii) is measurable, for a.e. . Evidently, is a Banach space endowed with the canonical norm (II) denotes the set of continuous random processes which satisfy(i), (ii) is measurable, for a.e. .(III) denotes the space of all valued -measurable random variables.(IV)For , is the space of all -measurable random variables such that .(V)For any , introduce the norm  on the Banach space For , all the norms with different are equivalent. is the Banach space endowed with the norm

The following result on BSEEs (see Lemma 2 in Mahmudov and McKibben [2]) will play a key role in proving the well-posedness of mean-field BSEEs.

Lemma 1. Let be a Hilbert space, and let be a linear operator which generates a -semigroup on . For any the following equation has a unique solution in ; moreover, where and is the space of bounded, linear operators on .

3. Mean-Field Backward Stochastic Evolution Equations

In this section, we study the existence and uniqueness result of mild solutions to mean-field BSEEs in a Hilbert space . To this end, we firstly recall some notations introduced by Buckdahn et al. [7].

Let be the (noncompleted) product of with itself and we define on this product space. A random variable originally defined on is extended canonically to . For any , the variable belongs to a.s., whose expectation is denoted by Note that and

The mean-field BSEE we consider has the following form: for any given measurable mapping and , where is the generator of a strongly continuous semigroup , , in the Hilbert space , with the notation .

Definition 2. We say that a pair of adapted processes is a mild solution of mean-field BSEE (11) if and for all

Remark 3. We emphasize that the coefficient of (11) can be interpreted as

3.1. Lipschitz Case

Now we study the existence and uniqueness of mild solutions to mean-field BSEE (11) under Lipschitz conditions. For , assume the following.(A1)There exists an such that  for all ,  , , , , .(A2).

We have the following theorem.

Theorem 4. For any random variable , under (A1) and (A2), mean-field BSEE (11) admits a unique mild solution .

Proof. Consider the following.
Step 1. For any , BSEE has a unique solution. In order to get this conclusion, we define Then (15) can be rewritten as Due to (A1), for all , , satisfies
According to Theorem 3.1 in [1], BSEE (15) has a unique solution.
Step 2. From Step 1, we can define a mapping through For any , we set , , , and . Then, from Lemma 1, we have If we set , then That is,
The estimate (22) shows that is a contraction on the space with the norm 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 mild solution of (11).

Arguing as the previous proof, we arrive at the following assertion in a straightforward way.

Corollary 5. Suppose that, for all in a metric space , is a given function satisfying (A1) and (A2) with independent on . Also suppose that in as for all .
If we denote by the mild solution of (11) corresponding to the functions and to the final data , then the map is continuous from to .

3.2. Non-Lipschitz Case

This subsection is devoted to finding some weaker conditions than the Lipschitz one under which the mean-field BSEE has a unique solution. To state our main result in this section, we suppose the following.

(A3) For all , , , , there exists an , such that where is a concave increasing function such that , for and .

In Mao [5], the author gave three examples of the function to show the generality of condition (A3). From these examples, we can see that Lipschitz condition (A1) is a special case of the given condition (A3).

Since is concave and , there exists a pair of positive constants and such that for all . Therefore, under assumptions (A2) and (A3), whenever and .

By Picard-type iteration, we now construct an approximate sequence, using which we obtain the desired result. Let , and, for , let be a sequence in defined recursively by on . From Theorem 4, (27) has a unique mild solution .

In order to give the main result, we need to prepare the following lemmas about the properties of , .

Lemma 6. Under hypotheses (A2) and (A3), there exist positive constants and such that for all and .

Proof. Using the hypotheses (A2) and (A3) with yields Then, it follows from Lemma 1 that where
If we set , we can obtain An application of the Gronwall inequality now implies Point (i) of Lemma 6 is now proved.
From formula (32), we know that This proves point (ii) of the Lemma.
To prove point (iii), we note that By Lemma 1 we have We can choose sufficiently large such that
Then where we set .

We divide the interval into subintervals by setting , with .

Lemma 7. For all , define Then, for all , the following inequality holds for a suitable :

Proof. Firstly, it needs to be verified that for all the following inequality holds provided is chosen sufficiently small.
Actually, this inequality holds provided that or Since , from , the above inequality holds if Thus, (41) holds for any , if . Therefore, we can choose a sufficiently large such that . Clearly, such a only depends on , , , , and .
Now, assume that (40) holds for some . Then, we have This completes the proof.

Now, we can give the main result of this section.

Theorem 8. Assume that (A2) and (A3) hold. Then, there exists a unique mild solution to (11).

Proof. Consider the following.
Uniqueness. To show the uniqueness, let both and be solutions of (11). For any , similar to the proof of (36), one can obtain
That is, if is sufficiently large, An application of Bihari inequality yields So for all a.s. It then follows from (46) that for all a.s. as well. This establishes the uniqueness.
Existence. We claim that the sequence defined by (27) satisfies as .
Indeed, for all , we set . By Lemmas 6 and 7, Suppose that holds for some . According to Lemma 6(iii) and Lemma 7, for all , we obtain This implies that, for all ,
By definition, is continuous on . Note that for each , is decreasing on , and for each , , is a nonincreasing sequence. Therefore, we define the function by . It is easy to verify that is continuous and nonincreasing on . By the definitions of and we get for all . Since , the Bihari inequality implies
For each , (52) and (54) yield Then, as , and this proves the assertion (49).
By (36), we obtain Applying (49) to the above formula, we see that is a Cauchy (hence convergent) sequence in ; denote the limit by . Now letting in (27), we obtain that holds on the entire interval . The theorem is now proved.

To illustrate the application of the obtained existence and uniqueness result, we consider the example of backward stochastic partial differential equations (BSPDEs) of mean-field type.

Example 9. Let be an open bounded domain in with uniformly boundary , let be a standard -dimensional Brownian motion (equipped with the normal filtration), and let be an -measurable random variable. We also let denote the semielliptic partial differential operator on of the form The aim is to study the solvability of the following initial boundary value problem: where The following assumptions will have to be in force.(H1), are uniformly continuous and bounded and satisfy the usual uniform ellipticity condition: , for some and all , .(H2) is measurable in and continuous in , and there exists such that for all , , , , , , , , , .
Then, we are now in a position of showing existence and uniqueness of the solution of BSPDEs (60).

Theorem 10. If (H1) and (H2) are satisfied, then the mean-field BSPDE (60) has a unique mild solution .

Proof. Let and . Define the operator by It is shown in [17] (see Example 2.1 in [17]) that generates a strongly continuous semigroup on . Define the maps by for all , . With these identifications, (60) can be written in the form of (11). By (H2), we know satisfy condition (A1). Hence, an application of Theorem 4 concludes that (60) has a unique mild solution .

4. Mean-Field Stochastic Evolution Equations

Let , , be a cylindrical Wiener process with values in a Hilbert space , defined on a probability space . We fix an interval and consider the stochastic evolution equations of mean-field type for an unknown process , with values in a Hilbert space : where operator is the generator of a strongly continuous semigroup , , in the Hilbert space , with .

By a mild solution of (65) we mean an -measurable process , , with continuous paths in , such that, -a.s.,

We suppose the following.

(A4) is a measurable mapping which satisfies for some constant .

(A5) The mapping fulfills that for every the map is measurable, for every , , , , , , , and for some constants and .

Theorem 11. Under assumptions (A3) and (A4), (65) has a unique mild solution .

The proof is constructed in two steps like that of Theorem 4 and it uses standard arguments for stochastic evolution equations introduced in the proof of Proposition  3.2 in [3]. Since the proof is straightforward, we prefer to omit it.

Remark 12. In our paper, Lipchitz condtion (A4) is given to get the well-posedness of mean-field stochastic evolution equations. In fact, (A4) can be replaced by a weaker condition such as (A3). We just give the condition (A4) for simplicity.

From standard arguments, we can also get the following continuous dependence theorem.

Corollary 13. Assume that for all in a metric space , satisfy (A4) and (A5) with and independent of  . Also assume that as for all .
If we denote by the mild solution of mean-field SEE (65) corresponding to the functions and to the initial data , then we have

5. Maximum Principle for BSPDEs of Mean-Field Type

5.1. Formulation of the Problem

Let be a bounded open set with smooth boundary and let , the space of controls, be a separable real Hilbert space. We denote An element of is called an admissible control.

For any , we consider the following controlled BSPDE system in the state space (norm , scalar product ): where is a partial differential operator, , and .

The cost functional is given by where Our purpose is to minimize the functional over , subject to the following state constraint: where

An admissible control that satisfies is called optimal.

Through what follows, the following assumptions will be in force.(L1) is a partial differential operator with appropriate boundary conditions. We assume that is the infinitesimal generator of a strongly continuous semigroup , in . Moreover, for every , for some constant independent of and .(L2), , , and are continuously Gâteaux differentiable with respect to . is continuously Gâteaux differentiable with respect to and is continuous with respect to .(L3)The derivatives of , , , and are Lipschitz continuous and bounded by  where is a positive constant.

Obviously, according to Theorem 4, state equation (72) has a unique mild solution under the above assumptions.

Remark 14. We can define the second order differential operator: By Example 9, fulfills assumption (L1) if , satisfy condition (H1).

5.2. Variation of the Trajectory

Let be an optimal control with being the corresponding optimal state. Let and . For any given , we introduce the spike variation of the control : It is clear that .

Let be the trajectory corresponding to . We use the following short notation for brevity:

Consider the following equation: Since the coefficients in (82) are bounded, it is easy to check that there exists a unique mild solution such that

We have the following estimate.

Theorem 15. There holds

Proof. We define For simplicity, let us define
By the definition of , , and , is the mild solution of with where we denote
For any , according to Lemma 1, we obtain
By condition (L3), we have
Combined with (91), (90) yields
We claim that From (89) where Then, Take ; for example, Note that The inequality above holds due to the boundedness of . Indeed, Assumption (L3) implies the boundedness of . Meanwhile is the solution of mean-field BSEE (82). It can be easy to check is bounded since the coefficients in (82) are bounded.
On the other hand, where is the mild solution of the following equation: and is the mild solution of
By the definition of , according to (L2), we have in as . Using the continuous dependence theorem Corollary 5, we obtain Then,
Combining (98) with (104), we finally have , as .
The required result (93) follows by using the similar estimations for , , and .
Note that if is sufficiently large. Now to prove the desired result (84) it suffices to apply Gronwall’s lemma and estimate (93) to inequality (92).

To deal with the state constraint (75), we need to recall the Ekeland variational principle.

Lemma 16 (Ekeland’s variational principle, see [16, Lemma 4.1]). Let be a complete metric space and let be lower semicontinuous and bounded from below. If, for , there exists , such that then there exists , satisfying

Now fix , and set where denotes the Lebesgue measure on .

The following result is proved as Proposition  4.1 in [16].

Lemma 17. is a complete metric space and is continuous and bounded on , where and is the mild solution of (72) corresponding to the control .

Now we consider the following free initial state optimal control problem: It is easy to check that According to Ekeland’s variational principle, there exists a such that Using the spike variation method, we can construct as follows: It is clear that . Let (resp., ) be the solution of (72) with respect to the control (resp., ). Following (82), is the mild solution of By Theorem 15, we know that

The proof of the following proposition is technical but based on the arguments above and we omit it.

Proposition 18. One has where

5.3. Variational Inequality and Adjoint Equation

In this subsection, the adjoint process is introduced to deduce the variational inequality.

If we set in (111) and notice that , we get By Lemma 17, where we set and use the limit as according to (115).

As for all , we know that there exists a subsequence of (still denoted by ) such that

Combining (115), (117) with (118), we get

Next, we introduce the adjoint equation corresponding to variational equation (113), whose solution is denoted by : where is the -adjoint operator of . Under assumptions (L1)–(L3), this is a linear mean-field SEE with bounded coefficients. An application of Theorem 11 implies that it has a unique adapted mild solution such that .

When , according to Corollaries 5 and 13, converges to , where is the solution of the following equation:

The following proposition, which formally follows from Proposition 18, gives the relation between and .

Proposition 19. Consider the following:

The following theorem constitutes the main contribution of this section, the maximum principle for the BSPDE control system.

Theorem 20. Let assumptions (L1)–(L3) hold. Suppose is an optimal control and is the corresponding optimal state trajectory for the BSPDE control systems (72) and (73) with the initial state constraint (75). Then there exists which satisfies (125), such that where is the Hamiltonian function defined by

Proof. By (122) and Proposition 19, we obtain
Letting in (129), we derive, for a.e. , for all .
Finally, taking in (130), we derive the desired result.

Remark 21. We note that if the coefficients do not depend explicitly on the marginal law of the underlying diffusion, the result reduces to the classical case, that is, the SMP for BSPDEs without mean-field term.

Remark 22. When we remove the initial state constraint (75), we obtain the general maximum principle for the mean-field BSPDEs system (i.e., without the constraint) with .

5.4. Application: A Backward Linear Quadratic Control Problem

Now, we apply our maximum principle to solve an LQ problem. For notational simplicity, we restrict ourselves to the free case (i.e., without the initial state constraint (75)), the general case being handled in a similar way.

Consider the following problem: subject to where is a partial differential operator satisfying condition (L1) and , , , , , and are bounded and deterministic constants. We also assume that and . It is easy to verify that BSPDE (132) admits a unique mild solution .

, the adjoint process of state equation (132), is the solution of Let be an optimal control, and let be the corresponding state process. By maximum principle of Theorem 20 (note that in this problem), for all since the state equation has the form (132). This in turn implies It is clear that (131) is a positive quadratic functional of control because of . Hence, an optimal control exists. The candidate optimal control (135) is indeed an optimal control of this LQ problem for it is the only control which satisfies the maximum principle.

Next, we want to obtain a more explicit representation of the optimal control (135) from the state equation (132). Substituting (135) into state equation (132) yields

Combining the above equation with (133), we obtain the following related feedback control system:

Looking at the terminal condition of in (133) and considering the mean-field type of (132), it is reasonable to conjecture that has the following form: where , are deterministic differential functions which will be specified below. Moreover, , .

Inserting this form into adjoint equation (133) and noticing that satisfies (136), we can compare the coefficients of and to obtain the following equation: Then, subtracting we have

Comparing the coefficients of and , respectively, we get where .

We solve (141) to get Then (142) exists, a unique solution from the classical Riccati equation theory.

We now conclude the above discussions in the following result.

Theorem 23. For one’s linear quadratic stochastic partial differential control problem (131)-(132), the unique optimal control is given by with satisfying (143) and solving (142).

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is partially supported by the Natural Science Foundation of China under Grant no. 11301310.