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First Passage Optimality and Variance Minimisation of Markov Decision Processes with Varying Discount Factors

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  30 January 2018

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

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This paper deals with the first passage optimality and variance minimisation problems of discrete-time Markov decision processes (MDPs) with varying discount factors and unbounded rewards/costs. First, under suitable conditions slightly weaker than those in the previous literature on the standard (infinite horizon) discounted MDPs, we establish the existence and characterisation of the first passage expected-optimal stationary policies. Second, to further distinguish the expected-optimal stationary policies, we introduce the variance minimisation problem, prove that it is equivalent to a new first passage optimality problem of MDPs, and, thus, show the existence of a variance-optimal policy that minimises the variance over the set of all first passage expected-optimal stationary policies. Finally, we use a computable example to illustrate our main results and also to show the difference between the first passage optimality here and the standard discount optimality of MDPs in the previous literature.

Keywords

Discrete-time Markov decision processvarying discount factorunbounded rewardfirst passage optimalityvariance minimisation

MSC classification

Primary: 90C40: Markov and semi-Markov decision processes
Secondary: 93E20: Optimal stochastic control 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 441 - 456
Copyright
© Applied Probability Trust 

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