Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper, we consider an optimal control problem with state constraints, where the control system is described by a mean-field forward-backward stochastic differential equation (MFFBSDE, for short) and the admissible control is mean-field type. Making full use of the backward stochastic differential equation theory, we transform the original control system into an equivalent backward form, i.e., the equations in the control system are all backward. In addition, Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control. We also study a stochastic linear quadratic control problem with state constraints.