Necessary and sufficient optimality conditions for relaxed and strict control problems of forward-backward systems

We consider a stochastic control problem of nonlinear forward-backward systems, where the set of strict (classical) controls need not be convex and the coefficients depend explicitly on the variable control. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality, in the form of global stochastic maximum principle, for two models. The first concerns the relaxed controls, who are a measure-valued processes. The second is a restriction of the first to strict control problems. AMS Subject Classification. 93 Exx