FBSDEs under Nonlinear Domination-Monotonicity Conditions and Optimal Controls of Linear SDEs

In this paper, by introducing nonlinear adjoint functions, the domination-monotonicity conditions that ensure the well-posedness of forward-backward stochastic differential equations (FBSDEs) are generalized from the existing linear case to the new nonlinear case. Applying the obtained well-posedness of FBSDEs and combining some other delicate techniques, we solve a linear-convex stochastic optimal control problem and a linear-quadratic (LQ) stochastic optimal control problem with input constraints where the constrained set can be time-dependent and random. Specifically, the existence and uniqueness of the optimal control for both problems, as well as their closed-form representations, are established.