Stochastic optimal control of McKean-Vlasov equations with anticipating law

In this paper, we generalise Pontryagin's stochastic maximum principle to controlled McKean-Vlasov equations with anticipating law. The associated new type of delayed backward equations with implicit terminal condition is studied.


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 [4], and Bensoussan [2], [3] and so on. Recently, this principle was extended to more general cases such as controlled MacKean-Vlasov systems by Carmona and Delarue [7], [8], in the sense that both the drift and the diffusion coefficients in their dynamics, are itself. 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 of mean-field systems has been studied for example by Anderson and Djehiche in [1] and Hu el al in [12] and for more details about mean-field systems, we refer to Lions [13], Cardaliaguet notes [6] and Buckdahn et al [5]. Delayed BSDEs have been studied by Delong and Imkeller [9]; they have later been extended by the same authors to the jump case, and studied [10] by the help of the Malliavin calculus and for more details, we refer to Delong's book [11].
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 section 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.

Solvability of the anticipated forward McKean-Vlasov equations
Let us consider the following anticipated SDE for a given positive constant δ    dX(t) = σ t, X(t), P X(t+δ) dB(t) + b t, X(t), P X(t+δ) dt, t ∈ [0, T ] , are progressively measurable and are assumed to satisfy the following set of assumptions.

For all
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.
Proposition 3.2 Under the above assumption (H.1), there is some δ 0 > 0, such that for all δ ∈ (0, δ 0 ], there exists a unique solution X ∈ S 2 F ([0, T ]) of SDE (3.1); δ 0 depends only on the Lipschitz constant C of the coefficients b and σ (see (H.1)) but not on the coefficients themselves.
This proves that Φ : Hence, there is a unique fixed point X ∈ H, such that X = Φ (X) , i.e., For a v ⊗ P −modification of X, also denoted by X, we have X ∈ S 2 F ([0, T ]) 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 This shows that, for every u ∈ U := L 0 F ([0, T ] ; U) and ω ∈ Ω, the coefficients σ and b satisfy the assumptions (H.1). Thus, for δ 0 > 0 from Proposition 3.2, for all u ∈ U;

Cost functional
Let us endow our control problem with a terminal cost g : and a running cost l :

Assumptions (H.3)
We suppose that g : [0, T ] × Ω × R d × P 2 R d → R is continuous and satisfies a linear growth assumption: For some constant C > 0, For any admissible control u, we define the performance functional: A control process u * ∈ U is called optimal, if J (u * ) ≤ J (u) , for all u ∈ U.
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.
Given an arbitrary but fixed control u ∈ U, we define Note that, thanks to the convexity of U and U, also u θ ∈ U, θ ∈ [0, 1]. We denote by X θ := X u θ and by X * := X u * the solution processes corresponding to u θ and u * , respectively. For simplicity of the computations, we set d = 1.
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 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 : for all u ∈ U. Assume for some u ∈ U, But this is a contradiction and proves that dt dP -a.e, for all u ∈ U. By the definition of H in (4.20), the proof is complete.
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.