Necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints

The main purpose of this work is to establish some first- and second-order necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints. The control may affect both the drift and the diffusion terms of the systems, and the control regions are allowed to be nonconvex. A stochastic inward pointing condition is proposed to ensure the normality of the corresponding necessary conditions.