Note on discounted continuous-time Markov decision processes with a lower bounding function

TL;DR

This paper studies discounted continuous-time Markov decision processes with a lower bounding function, establishing conditions for the existence of stationary optimal policies and introducing a novel transformation technique.

Contribution

It introduces a new transformation for nonhomogeneous Markov jump processes to analyze CTMDPs, allowing weaker conditions for optimal policy existence.

Findings

Established existence of stationary optimal policies under weaker conditions.

Applied a novel transformation technique to analyze CTMDPs.

Extended the theoretical framework for discounted CTMDPs with unbounded costs.

Abstract

In this paper, we consider the discounted continuous-time Markov decision process (CTMDP) with a lower bounding function. In this model, the negative part of each cost rate is bounded by the drift function, say $w$, whereas the positive part is allowed to be arbitrarily unbounded. Our focus is on the existence of a stationary optimal policy for the discounted CTMDP problems out of the more general class. Both constrained and unconstrained problems are considered. Our investigations are based on a useful transformation for nonhomogeneous Markov pure jump processes that has not yet been widely applied to the study of CTMDPs. This technique was not employed in previous literature, but it clarifies the roles of the imposed conditions in a rather transparent way. As a consequence, we withdraw and weaken several conditions commonly imposed in the literature.