Small noise perturbations of stochastic ergodic control problems

TL;DR

This paper investigates the effects of small noise perturbations on stochastic ergodic control problems, providing error bounds and a novel representation of the ergodic value through tunneling and Markov chains.

Contribution

It introduces a small noise limit approach to analyze degenerate stochastic ergodic control problems and establishes a tunneling method for a specific case, offering new insights and representations.

Findings

Derived error bounds for ε-optimal controls.

Established tunneling phenomena in a special ergodic control problem.

Provided a new representation of the ergodic value using tunneled Markov chains.

Abstract

Using small noise limit approach, we study degenerate stochastic ergodic control problems and as a byproduct obtain error bounds for the $\varepsilon$-optimal controls. We also establish tunneling for a special ergodic control problem and give a representation of the ergodic value using the tunneled Markov chain.