Impulse control maximising average cost per unit time: a non-uniformly ergodic case

TL;DR

This paper investigates impulse control strategies to maximize average cost per unit time for ergodic Feller-Markov processes, allowing for non-uniform convergence and unbounded impulse costs, and establishes the existence of optimal strategies.

Contribution

It extends existing results by handling non-uniform ergodicity and unbounded costs, providing a broader framework for impulse control problems.

Findings

Optimal value is independent of initial state.

Existence of optimal or near-optimal strategies.

Handles non-uniform convergence and unbounded costs.

Abstract

This paper studies maximisation of an average-cost-per-unit-time ergodic functional over impulse strategies controlling a Feller-Markov process. The uncontrolled process is assumed to be ergodic but, unlike the extant literature, the convergence to invariant measure does not have to be uniformly geometric in total variation norm; in particular, we allow for non-uniform geometric or polynomial convergence. Cost of an impulse may be unbounded, e.g., proportional to the distance the process is shifted. We show that the optimal value does not depend on the initial state and provide optimal or $\ve$-optimal strategies.