A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first- and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear backward stochastic differential equations (BSDEs) with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales and prove the solvability for a class of multidimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.