  Stochastic maximum principle of nonlinear controlled forward-backward
systems, where the set of strict (classical) controls need not be convex and
the diffusion coefficient depends explicitly on the variable control, is an
open problem impossible to solve by the classical method of spike variation. In
this paper, we introduce a new approach to solve this open problem and 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.
