Numerical Method for Ergodic Optimal Control Problems of Switching Stochastic Differential Equations with Reflection

Abstract

The ergodic or long run average cost control problem for diffusions is one of the few classical problems of stochastic control that still eludes a completely satisfactory treatment. This is certainly true for the setting in which the systems to be controlled is modeled by the solution of a switching stochastic differential equation with reflection (SSDER). In this paper we set forth a preliminary study for this setting in the unidimensional case. Besides carving out a numerical method, our treatment of the ergodic control in this scenario straddles issues of existence and uniqueness of solution of the the SSDER and a verification theorem for the associated HJB equation. We conclude by illustrating the effectiveness of the method considering the control energy consumption in a large parallel processing computer system composed of one queue and several processing stations.