Ergodic control for constrained diffusions: characterization using HJB equations

Recently in [A. Budhiraja, SIAM J. Control Optim., 42 (2003), pp. 532--558] 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 multivalued 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 precompactness of the family of suitably renormalized discounted value functions. In this regard we use a recent technique, introduced in [V. S. Borkar, Stochastic Process Appl., 103 (2003), pp. 293--310], of using the Athreya--Ney--Nummelin pseudoatom construction for obtaining a coupling of a pair of embedded, discrete time, controlled Markov chains