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Stochastic maximum principle of nonlinear controlled forward-backwardsystems, where the set of strict (classical) controls need not be convex andthe diffusion coefficient depends explicitly on the variable control, is anopen problem impossible to solve by the classical method of spike variation. Inthis paper, we introduce a new approach to solve this open problem and weestablish necessary as well as sufficient conditions of optimality, in the formof global stochastic maximum principle, for two models. The first concerns therelaxed controls, who are a measure-valued processes. The second is arestriction of the first to strict control problems.

A general stochastic maximum principle for optimal control problems of forward-backward systems