A Global Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Systems

We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear forward-backward stochastic differential equations. Inspired by Hu [Probab. Uncertain. Quant. Risk, 2 (2017), pp. 1--20], we develop a new decoupling approach by introducing an adjoint equation which is a quadratic backward stochastic differential equation. By revealing the relations among the terms of the first-order Taylor expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated.