Stochastic Maximum Principle for Optimal Control Problems of Forward-Backward Systems Involving Impulse Controls

We consider a stochastic optimal control problem of a forward-backward system in which the control variable consists of two components: the continuous control and the impulse control. The domain of the control is assumed to be convex. Necessary optimality conditions of the Pontryagin maximum principle type are obtained for this stochastic optimal control problem. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient.