Stochastic Perron's method for optimal control problems with state constraints

We apply the stochastic Perron method of Bayraktar and Sirbu to a general infinite horizon optimal control problem, where the state X is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function v is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth domain, under some additional assumptions, these estimates allow to identify v with a unique continuous constrained viscosity solution of this equation.