The Maximum Principle For Optimal Control Problem Of Mean-Field Forward-Backward Stochastic System With State Constrained

We study a stochastic optimal control problem where the controlled sys-tem are described by decoupled mean-field forward-backward stochastic e-quation (mean-field FBSDE) and fully coupled mean-field forward-backward stochastic equation, respectively, while the forward state is constrained in a closed convex set at. the terminal time. We make full use of the theo-ry of forward-backward stochastic differential equations, mean-field forward-backward stochastic differential equations and the optimal control, etc.Firstly, we study a stochastic optimal control problem where the controlled system is described by decoupled mean-field forward-backward stochastic e-quation, while the forward state is constrained in a closed convex set at the terminal time. That is, we consider the following mean-field forward-backward SDE:where the terminal state x(T) of the forward equation is constrained in a closed convex set.Then, we consider the following cost functional:We assume:(1) b, σ, g, h, l, φ, 7 are continuous in their variables and continuously differentiable in (x, y, z, x, y, z, u)Respectively;(2) The derivatives of b, a, g, h in(x, y, z, x, y, z, u) are bounded;(3) The derivatives of Tin (x, y, z,￡,y, z,u) are bounded by C(l+|cc|+|y|+ |z|+|x|+|y|+|z|+|u|) and the derivatives of φ,γ are bounded by C(1+|x|+|x|).The aim of this paper is to obtain a characterization of x(T) when the cost functional J(?) attains the minimize. By introducing an equivalent backward control problem, we use Ekeland’s variational principle to obtain a stochastic maximum principle.We consider the following fully coupled mean-field FBSDE:where: θ(t) = (x(t),y(t), z(t), E[x(t)]t E[y(t)}, E[z(t)]). We assume the monotonic condition:The equation (2) exists a unique solution under the Lipschitz condition,integrability condition and monotonic condition.Then we study a stochastic optimal control problem where the controlled system is described by fully coupled mean-field forward-backward stochastic equation:where θ(t)=(x(t),y(t),z(t),E[x(t)],E[y(t)],E[z(t))), u(t)∈Uad, we require the terminal state x(T) of the forward equation to lie in a closed convex set. Using similar methods, we get a stochastic maximum principle.