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Equivalence of Lyapunov stability criteria in a class of Markov decision processes

Research paper by Rolando Cavazos-Cadena, Onésimo Hernández-Lerma

Indexed on: 01 Sep '92Published on: 01 Sep '92Published in: Applied Mathematics & Optimization



Abstract

We are concerned with Markov decision processes with countable state space and discrete-time parameter. The main structural restriction on the model is the following: under the action of any stationary policy the state space is acommunicating class. In this context, we prove the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function; these conditions include the Lyapunov function condition (LFC) introduced by A. Hordijk. As a consequence of our results, the LFC is proved to be equivalent to the following: under the action of any stationary policy the corresponding Markov chain has a unique invariant distribution which depends continuously on the stationary policy being used. A weak form of the latter condition was used by one of the authors to establish the existence of optimal stationary policies using an approach based on renewal theory.