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Finite-horizon optimality for continuous-time Markov decision processes with unbounded transition rates

Part of: Markov processes Stochastic systems and control

Published online by Cambridge University Press:  21 March 2016

Xianping Guo ,
Xiangxiang Huang  and
Yonghui Huang
Xianping Guo*
Affiliation:
Sun Yat-Sen University
Xiangxiang Huang*
Affiliation:
Sun Yat-Sen University
Yonghui Huang*
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.
∗ 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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In this paper we focus on the finite-horizon optimality for denumerable continuous-time Markov decision processes, in which the transition and reward/cost rates are allowed to be unbounded, and the optimality is over the class of all randomized history-dependent policies. Under mild reasonable conditions, we first establish the existence of a solution to the finite-horizon optimality equation by designing a technique of approximations from the bounded transition rates to unbounded ones. Then we prove the existence of ε (≥ 0)-optimal Markov policies and verify that the value function is the unique solution to the optimality equation by establishing the analog of the Itô-Dynkin formula. Finally, we provide an example in which the transition rates and the value function are all unbounded and, thus, obtain solutions to some of the unsolved problems by Yushkevich (1978).

Keywords

Continuous-time Markov decision processfinite-horizon criterionoptimal Markov policyrandomized history-dependent policyunbounded transition rate

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
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1064 - 1087
Copyright
Copyright © Applied Probability Trust 2015 

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