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

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.


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][2][3][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, meanfield 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][7][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.
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.

Preliminaries
The norm of an element  in a Banach space  is denoted by ||  or simply ||, if no confusion is possible.Γ, , and  are three real and separable Hilbert spaces.Scalar product is denoted by ⟨⋅, ⋅⟩, with a subscript to specify the space, if necessary.L(Γ, ) is the space of Hilbert-Schmidt operators from Γ to , endowed with the Hilbert-Schmidt norm.
By F  ,  ∈ [0, ], we denote the natural filtration of , augmented with the family N of P-null sets of F  : The filtration (F  ) ≥0 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 .
(V) For any  ∈ R, introduce the norm on the Banach space For 0 <  < ∞, all the norms ‖ ⋅ ‖ , with different  ∈ R are equivalent.K[0, ] = K 0 [0, ] 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.

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 .
Arguing as the previous proof, we arrive at the following assertion in a straightforward way.
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 (0) = 0, there exists a pair of positive constants  and  such that for all  ≥ 0. Therefore, under assumptions (A2) and By Picard-type iteration, we now construct an approximate sequence, using which we obtain the desired result.Let  0 () ≡ 0, and, for  ∈ N, let {  ,   } be a sequence in on 0 ≤  ≤ .From Theorem 4, ( 27) has a unique mild solution (  (),   ()).
Proof.Using the hypotheses (A2) and (A3) with () ≤  +  yields Then, it follows from Lemma 1 that where If we set  = 96 2  max{, }, 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.

Lemma 7. For all 𝑡 ∈ [𝜏
Then, for all  ≥ 1, the following inequality holds for a suitable  > 0: Proof.Firstly, it needs to be verified that for all  ∈ [ −1 ,   ] the following inequality holds provided  > 0 is chosen sufficiently small.Actually, this inequality holds provided that Since  1 > 1, from () ≤  + , the above inequality holds if This completes the proof.Now, we can give the main result of this section.
Proof.Consider the following.
Example 9. Let O be an open bounded domain in R  with uniformly  2 boundary O, let () be a standard dimensional Brownian motion (equipped with the normal filtration), and let  : O → R be an F  -measurable random variable.We also let  denote the semielliptic partial differential operator on  2 (R) 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.
(H2)  is measurable in (, , ỹ, z, , ) and continuous in (z, ), and there exists  > 0 such that      (, , ỹ1 , z1 , Then, we are now in a position of showing existence and uniqueness of the solution of BSPDEs (60).

Theorem 11. Under assumptions (A3) and (A4), (65) has a unique mild solution 𝑋(⋅) ∈ S 2 F ([𝑡, 𝑇]; 𝐾).
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  1 and  2 independent of .Also assume that as  →  0 for all  ∈ S 2 F ([0, ]; ).If we denote by   (⋅) the mild solution of mean-field SEE (65) corresponding to the functions (  ,   ) and to the initial data , then we have

Maximum Principle for BSPDEs of Mean-Field Type
An element of U is called an admissible control.
For any V ∈ U, we consider the following controlled BSPDE system in the state space  =  The cost functional is given by where Our purpose is to minimize the functional (⋅) over U ad , subject to the following state constraint: where An admissible control  ∈ U ad that satisfies 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   ,  ≥ 0 in .
where  is a positive constant.
Obviously, according to Theorem 4, state equation (72) has a unique mild solution under the above assumptions.
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.(84) Proof.We define For simplicity, let us define By the definition of (   ,    ), (  ,   ), and (   ,    ), (   ,    ) is the mild solution of with where we denote For any  > 0, according to Lemma 1, we obtain By condition (L3), we have Combined with (91), (90) yields We claim that From ( 89) where Then, Now fix V ∈ U ad , and set where  denotes the Lebesgue measure on R.
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 V.
Now we consider the following free initial state optimal control problem: inf 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: The proof of the following proposition is technical but based on the arguments above and we omit it.
Proposition 18.One has where where we set and use the limit as  → 0 according to (115). As Combining ( 115), ( 117) with (118), we get    The following theorem constitutes the main contribution of this section, the maximum principle for the BSPDE control system.
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  1 = 1.

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.