Stochastic optimal control of McKean–Vlasov equations with anticipating law

We are interested in Pontryagin’s stochastic maximum principle of controlled McKean–Vlasov stochastic differential equations. We allow the law to be anticipating, in the sense that, the coefficients (the drift and the diffusion coefficients) depend not only of the solution at the current time t, but also on the law of the future values of the solution PX(t+δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{X(t+\delta )}$$\end{document}, for a given positive constant δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}. We emphasise that being anticipating w.r.t. the law of the solution process does not mean being anticipative in the sense that it anticipates the driving Brownian motion. As an adjoint equation, a new type of delayed backward stochastic differential equations (BSDE) with implicit terminal condition is obtained. By using that the expectation of any random variable is a function of its law, our BSDE can be written in a simple form. Then, we prove existence and uniqueness of the solution of the delayed BSDE with implicit terminal value, i.e. with terminal value being a function of the law of the solution itself.


Introduction
The stochastic maximum principle and the characterization of the optimal control by Pontryagin's maximum principle have been studied for the classical case by many authors such as Bismut [9], and Bensoussan [8] and so on. This principle was extended to more general cases such as controlled MacKean-Vlasov systems in the sense that both the drift and the diffusion coefficients in their dynamics, are supposed to depend at time t on the solution X (t) and on its law P X (t) as follows In order to write it in a more comprehensible form, we use the fact that the expectation of any random variable is a function of its law, and under suitable assumptions on both the driver and a terminal value, we can get existence and uniqueness of our delayed BSDE. It is a generalisation of the adjoint equation of the above mentioned problem we are interested in. Stochastic Pontryagin's maximum principle in both cases partial and complete information, for a case where the mean-field term is expressed by the expected value of the state, has been studied for example by Anderson and Djehiche [7], Hu et al. [5,18] and Agram and Røse [6]. For more general mean-field problems, we refer to Lions [19], Cardaliaguet [12] and Buckdahn et al. [11]. Delayed BSDEs have been studied by Delong and Imkeller [15]; they have later been extended by the same authors to the jump case, and studied [16] by the help of the Malliavin calculus and for more details, we refer to Delong's book [17]. For mean-field delayed BSDE, we refer to Agram [1]. To the best of our knowledge our paper is the first to study optimal control problems of mean-field SDEs with anticipating law. The paper is organized as follows: in the next section, we give some preliminaries which will be used throughout this work. In Sect. 3, the existence and the uniqueness of McKean-Vlasov SDEs with anticipating law is investigated. Section 4 is devoted to the study of Pontryagin's stochastic maximum principle. In the last section, we prove the existence and the uniqueness for the associated delayed McKean-Vlasov BSDEs with implicit terminal condition.
This work has been presented at seminars and conferences in Brest, Biskra, Marrakech, Mans and Oslo.

Framework
We introduce some notations, definitions and spaces which will be used throughout this work. Let ( , F , P) be a complete probability space, B a d-dimensional Brownian motion and F = (F t ) t≥0 the Brownian filtration generated by B and completed by all P-null sets. Let P 2 (R d ) := {μ ∈ P(R d ) : R d |x| 2 μ(dx) < +∞}, where P(R d ) is the space of all the probability measures on (R d , B(R d )); recall that B(R d ) denotes the Borel σ -field over R d . We endow P 2 (R d ) with the 2-Wasserstein metric W 2 on P 2 (R d ): For μ 1 , μ 2 ∈ P 2 (R d ), the 2-Wasserstein distance is defined by We also remark that, if ( , F , P) is "rich enough" in the sense that then we also have Let (˜ ,F,P) := ( , F , P),F := F and (¯ ,F,P) = ( , F , P) ⊗ (˜ ,F,P). For any measurable space (E, E) and any random variable ζ : We observe that ζ on (˜ ,F,P) is a copy of ζ on ( , F , P), and ζ ,ζ are i.i.d underP. Moreover, for ζ , η : ( , F , P) → (E, E) random variables and ϕ : (E 2 , E 2 ) → (B, B(R)) a bounded and measurable function, we havẽ We recall now the notion of derivative of a function ϕ : P 2 (R d ) → R w.r.t a probability measure μ, which was studied by Lions in his course at Collège de France in [19]; see also the notes of Cardaliaguet [12], the works by Carmona and Delarue [14] and in Buckdahn et al. [11]. We say that ϕ is differentiable at μ if, for the lifted functionφ(ζ ) := ϕ(P ζ ), ζ ∈ L 2 , F , P; R d , there is some ζ 0 ∈ L 2 , F , P; R d with P ζ 0 = μ, such thatφ is differentiable in the Frèchet sense at ζ 0 , such that there exists a linear continuous mapping given by Riesz' representation theorem, we can write In Lions [19] and Cardaliaguet [12], it has been proved that there exists a Borel function h : R → R, such that (Dφ)(ζ 0 ) = h(ζ 0 ) P-a.s. Note that h(ζ 0 ) P-a.s. uniquely determined. Consequently, h(y) is P ζ 0 (dy)-a.e. uniquely determined. We define Throughout this work, we will use also the following spaces: is the set of real valued F-adapted continuous processes (X (t)) t∈[0,T ] such that is the set of real valued square integrable F t -measurable random variables.

Solvability of the anticipated forward McKean-Vlasov equations
Let us consider the following anticipated SDE for a given positive constant δ where P 0 is the distribution law of zero, i.e., the Dirac measure with mass at zero.
and a similar estimate holds for b.  Then V ∈ S 2 F ([0, T ]) ⊂ H (with the above identification), and setting (U ) := V we define a mapping : H → H . Fixing β > 0 (β will be specified later), we introduce the norm Obviously, (H , · −β ) is a Banach space, and the norm · −β is equivalent to the norm · 0 (obtained from · −β by taking β = 0). We are going to prove that Indeed, we recall that Hence for t = T , we have We seek suitable β > 0, δ > 0 with δ ≤ 1 β , i.e., βδ ≤ 1, in order to estimate i.e., This proves that and

Pontryagin's stochastic maximum principle
Let us introduce now our stochastic control problem.

Controlled stochastic differential equation
As control state space we consider a bounded convex subset is the set of all admissible controls. The dynamics of our controlled system are driven by functions σ :

Assumptions (H.2):
The coefficients σ and b are supposed to be continuous on On the other hand, from the continuity of the coefficients on This shows that, for every u ∈ U : and ω ∈ , the coefficients σ and b satisfy the Assumptions (H.1). Thus, for δ 0 > 0 from Proposition 3.2, for all u

Cost functional
Let us endow our control problem with a terminal cost g : and satisfies a linear growth assumption: For some constant C > 0, For any admissible control u, we define the performance functional: Let us suppose that there is an optimal control u * ∈ U. Our objective is to characterise the optimal control. For this let us assume some additional assumptions.

Variational SDE
Given u * ∈ U and the associated controlled state process From the previous section, we have the existence and the uniqueness of a solution for all δ ∈ 0, δ 0 ; 0 < δ 0 ≤ δ 0 small enough. Indeed, Eq. (4.1) is of the form where α i : [0, T ] × → R, i = 1, 2, 3, 4, are bounded progressively measurable processes and β j : [0, T ] × → R, j = 1, 2, two bounded (F t ⊗F T )progressively measurable processes. With the method used in the proof of Proposition 3.2, we get the existence of δ 0 ∈ (0, δ 0 ] stated above. It turns out that Y (t) is the L 2 -derivative of X θ (t) w.r.t. θ at θ = 0. More precisely, the following property holds.

Proof
The proof is obtained with standard computations. For the sake of completeness, we give details in the Appendix.

Adjoint processes
Let us first recall the equation satisfied by the derivative process where for notational convenient, we have used the short hand notations E ∂ μ σ (t, X * (t) , P X * (t+δ) ,X * (t + δ) , u * (t))Ỹ (t + δ) =:Ẽ ∂ μ σ (t)Ỹ (t + δ) , and similarly.In order to determine the adjoint backward equation, we suppose that it has the form

Stochastic maximum principle
We define now the Hamiltonian H : is the solution of the adjoint equation (4.18), (4.19).
Proof From (4.15) and (4.3) with the choice (4.16) and (4.17), we get for all u ∈ U. Assume for some u ∈ U, But this is a contradiction and proves that dt d P-a.e, for all u ∈ U. By the definition of H in (4.20), the proof is complete.

Solvability of the delayed McKean-Vlasov BSDE
We now study the BSDE which is the adjoint equation to the above control problem. We consider the BSDE which we have seen due to our computations that it has the form dp(t) = − (∂ x b)(t, X * (t) , P X * (t+δ) , u * (t)) p(t) with Let us better understand the form of this BSDE: and in order to describe also the terminal condition of our BSDE, we consider the coefficient We know that , ω ∈ , for ζ ∈ L 2 ¯ ,F,P .

Remark 5.2
We call the BSDE (5.4), delayed BSDE because the driver at time t depend on both the solution at time t and on its previous value, i.e. the solution at time t − δ.

Remark 5.3
The adjoint BSDE we describe it above is a special case of (5.5). Indeed, for the adjoint BSDE we have: and if (5.5) is satisfied.