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

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.