  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 FBSDEs. Inspired by Hu (Hu, Probability, Uncertainty and
Quantitative Risk, 2(1) (2017):pp 1-20), we develop a new decoupling approach
by introducing an adjoint equation which is a quadratic BSDE. By revealing the
relations among the terms of the first-order Taylor's 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.
