Ergodic Risk-Sensitive Control of Markov Processes on Countable State Space Revisited

TL;DR

This paper thoroughly analyzes ergodic risk-sensitive control problems for Markov processes on countable spaces, establishing existence, uniqueness, and optimality conditions, and revisiting stability assumptions with policy improvement algorithms.

Contribution

It provides a comprehensive analysis of ergodic risk-sensitive control for Markov processes, including new existence, uniqueness, and verification results under different conditions.

Findings

Established uniqueness of the value function

Proved existence of optimal stationary controls

Developed policy improvement algorithms

Abstract

We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.