Some new results on sample path optimality in ergodic control of diffusions

TL;DR

This paper improves conditions for sample path optimality in ergodic control of diffusions, weakening the moment requirements and providing new sufficient conditions for the existence of optimal controls.

Contribution

It introduces a weaker moment condition involving logarithmic terms and establishes new Foster-Lyapunov criteria for ergodic control optimality.

Findings

Weaker moment condition: ${ m E}[ au\,\ln^+( au)]$ instead of ${ m E}[ au^2]$.

New Foster-Lyapunov conditions for sample path optimality.

Applicability to models lacking uniform stability or near-monotone costs.

Abstract

We present some new results on sample path optimality for the ergodic control problem of a class of non-degenerate diffusions controlled through the drift. The hypothesis most often used in the literature to ensure the existence of an a.s. sample path optimal stationary Markov control requires finite second moments of the first hitting times $\tau$ of bounded domains over all admissible controls. We show that this can be considerably weakened: ${\mathrm E}[\tau^2]$ may be replaced with ${\mathrm E}[\tau\ln^+(\tau)]$, thus reducing the required rate of convergence of averages from polynomial to logarithmic. A Foster-Lyapunov condition which 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.