The Pontryagin's maximum principle for stochastic differential systems under state constraints

In this study, we consider an optimal control problem driven by a stochastic differential equation with state constraints. Here, the state constraints mean the constraints about the path of state. In order to show the maximum principe for the optimal control problem under state constraints, we investigate a new near optimal control problem. In the following, we establish the stochastic maximum principle for the new optimal control problem under multi-time state constraints. In the end, we give a production planning example to verify the main results of this study.