Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control Problems

We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear stochastic differential equation, in which the control enters both the drift and the diffusion coefficients. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are a measure-valued processes in which an optimal solution exists. The second is a particular case of the first and relates to strict control problems. These results are given in the form of global stochastic maximum principle by using only the first order expansion and the associated adjoint equation. This improves all the previous works on the subject. AMS Subject Classification. 93 Exx