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New discount and average optimality conditions for continuous-time Markov decision processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Xianping Guo  and
Liuer Ye
Xianping Guo*
Affiliation:
Sun Yat-Sen University
Liuer Ye*
Affiliation:
Sun Yat-Sen University
*
∗ Postal address: School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, P. R. China.
∗ Postal address: School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, P. R. China.
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Abstract

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This paper deals with continuous-time Markov decision processes in Polish spaces, under the discounted and average cost criteria. All underlying Markov processes are determined by given transition rates which are allowed to be unbounded, and the costs are assumed to be bounded below. By introducing an occupation measure of a randomized Markov policy and analyzing properties of occupation measures, we first show that the family of all randomized stationary policies is ‘sufficient’ within the class of all randomized Markov policies. Then, under the semicontinuity and compactness conditions, we prove the existence of a discounted cost optimal stationary policy by providing a value iteration technique. Moreover, by developing a new average cost, minimum nonnegative solution method, we prove the existence of an average cost optimal stationary policy under some reasonably mild conditions. Finally, we use some examples to illustrate applications of our results. Except that the costs are assumed to be bounded below, the conditions for the existence of discounted cost (or average cost) optimal policies are much weaker than those in the previous literature, and the minimum nonnegative solution approach is new.

Keywords

Occupation measureoptimality inequality/equationdiscounted cost optimal policyaverage cost optimal policyminimum nonnegative solution method

MSC classification

Primary: 90C40: Markov and semi-Markov decision processes 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 953 - 985
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported by the NSFC and GDUPS (2010).

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