An Ergodic Control Problem for Constrained Diffusions
Amarjit Budhiraja
Department of Statistics
University of North Carolina - Chapel Hill
Abstract
In this work we consider an ergodic control problem for a class
of diffusion processes, constrained to take values in a
polyhedral cone. The goal is the almost sure minimization of
long term cost per unit time. We show 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. Once these properties
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 talk,
is proved. Next we characterize the value of the ergodic control
problem via a suitable Hamilton-Jacobi-Bellman (HJB) equation.
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 fucnction of the control
problem. The existence of a solution to this JHB 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 by Borkar, of using the
Athreya-Ney-Nummelin pseudo-atom construction for obtaining a
coupling of a pair of embedded, discrete time, controlled Markov
chains.