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This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations (which are quite different from the case of convex control constraint), we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of the Pontryagin-type maximum principle.

Pointwise Second-Order Necessary Conditions for Stochastic Optimal Controls, Part II: The General Case