Stochastic Controls with Terminal Contingent Conditions

This paper considers a nonlinear stochastic control problem where the system dynamics is a controlled nonlinear backward stochastic differential equation and the state must coincide with a given random vector at the terminal time. A necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented. The general result is also applied to backward linear-quadratic control problem and an optimal control is obtained explicitly as a feedback of the solution to a forward]backward equation. Finally, a nonlinear problem with additional integral constraints is discussed and it is shown that the duality gap is zero under the Slater condition