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Countable state Markov decision processes with unbounded jump rates and discounted cost: optimality equation and approximations

Part of: Markov processes Stochastic systems and control

Published online by Cambridge University Press:  21 March 2016

H. Blok  and
F. M. Spieksma
H. Blok*
Affiliation:
Leiden University
F. M. Spieksma*
Affiliation:
Leiden University
*
∗ Postal address: Mathematisch Instituut, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands.
∗ Postal address: Mathematisch Instituut, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands.
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Abstract

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This paper considers Markov decision processes (MDPs) with unbounded rates, as a function of state. We are especially interested in studying structural properties of optimal policies and the value function. A common method to derive such properties is by value iteration applied to the uniformised MDP. However, due to the unboundedness of the rates, uniformisation is not possible, and so value iteration cannot be applied in the way we need. To circumvent this, one can perturb the MDP. Then we need two results for the perturbed sequence of MDPs: 1. there exists a unique solution to the discounted cost optimality equation for each perturbation as well as for the original MDP; 2. if the perturbed sequence of MDPs converges in a suitable manner then the associated optimal policies and the value function should converge as well. We can model both the MDP and perturbed MDPs as a collection of parametrised Markov processes. Then both of the results above are essentially implied by certain continuity properties of the process as a function of the parameter. In this paper we deduce tight verifiable conditions that imply the necessary continuity properties. The most important of these conditions are drift conditions that are strongly related to nonexplosiveness.

Keywords

Parametrised Markov processesnonexplosivenessdiscounted costdrift conditionsperturbed MDPs

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. 1088 - 1107
Copyright
Copyright © Applied Probability Trust 2015 

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