A maximum principle for fully coupled forward–backward stochastic control systems with terminal state constraints

We study a stochastic optimal control problem where the controlled system is described by a fully coupled forward–backward stochastic differential equation (FBSDE), while the forward state is constrained in a convex set at the terminal time. By introducing an equivalent backward control problem, we use terminal variation approach to obtain a stochastic maximum principle. Applications to the utility optimization problem in the financial market and state constrained stochastic linear quadratic control models are investigated.