An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control

Abstract

An ergodic control problem for a class of constrained diffusion processes is considered. The goal is the almost sure minimization of long term cost per unit time. The main result of the paper is that there exists an optimal Markov control for the considered problem. It is shown that under the assumption of regularity of the Skorohod map and the assumption that the drift vector field takes values in a certain cone of stability, the class of controlled diffusion processes considered have strong, uniform in control, stability properties. The role of the boundary is critical in obtaining the stability and ergodic control results for the class of controlled constrained diffusion processes considered in this work since the domains are unbounded and the corresponding unconstrained diffusions are typically transient. These stability properties are key in obtaining appropriate tightness estimates. Once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures. In this regard an extension to the Echeverria--Weiss--Kurtz characterization of invariant measures of Markov processes, to the case of constrained-controlled processes considered in this paper, is proved. This characterization result is also crucially used in proving the compactness of the family of invariant measures of Markov processes corresponding to all possible Markov controls.