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

TL;DR

This paper extends the stochastic Perron method to infinite horizon optimal control problems with state constraints, establishing bounds and uniqueness of the value function as a viscosity solution of the Hamilton-Jacobi-Bellman equation.

Contribution

It adapts the stochastic Perron method to handle state constraints in infinite horizon control problems, providing new bounds and conditions for the value function's characterization.

Findings

Value function is bounded by viscosity solutions.

In smooth domains, the value function is uniquely identified as a continuous viscosity solution.

The method applies to general controlled diffusion processes with state constraints.

Abstract

We apply the stochastic Perron method of Bayraktar and S\^irbu 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.