Ergodic Control for Constrained Diffusions: Characterization Using HJB Equations

Recently in [8] an ergodic control problem for a class of diffusion processes, constrained to take values in a polyhedral cone, was considered. The main result of that paper was that under appropriate conditions on the model, there is a Markov control for which the infimum of the cost function is attained. In the current work we characterize the value of the ergodic control problem via a suitable Hamilton-Jacobi-Bellman (HJB) equation. The theory of existence and uniqueness of classical solutions, for PDEs in domains with corners and reflection fields which are oblique, discontinuous and multi-valued on corners, is not available. We show that the natural HJB equation for the ergodic control problem admits a unique continuous viscosity solution which enables us to characterize the value function of the control problem. The existence of a solution to this HJB equation is established via the classical vanishing discount argument. The key step is proving the pre-compactness of the family of suitably re-normalized discounted value functions. In this regard we use a recent technique, introduced in [4], of using the Athreya-Ney-Nummelin pseudo-atom construction for obtaining a coupling of a pair of embedded, discrete time, controlled Markov chains.