Some New Results on Sample Path Optimality in Ergodic Control of Diffusions

摘要

We present some new results on sample path optimality for the ergodic control problem of a class of nondegenerate diffusions controlled through the drift. The hypothesis most often used in the literature to ensure the existence of an almost sure sample path optimal stationary Markov control requires finite second moments of the first hitting times τ of bounded domains over all admissible controls. We show that this can be considerably weakened: E[τ] may be replaced with E[τ ln (τ)], thus reducing the required rate of convergence of averages from polynomial to logarithmic. A Foster-Lyapunov condition that guarantees this is also exhibited. Moreover, we study a large class of models that are neither uniformly stable nor have a near-monotone running cost, and we exhibit sufficient conditions for the existence of a sample path optimal stationary Markov control.