Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control Problems

Abstract

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, which are 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 of the previous works on the subject.