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Multiobjective Stopping Problem for Discrete-Time Markov Processes: Convex Analytic Approach

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

F. Dufour  and
A. B. Piunovskiy
F. Dufour*
Affiliation:
Université Bordeaux I
A. B. Piunovskiy*
Affiliation:
University of Liverpool
*
∗Postal address: Université Bordeaux I and INRIA Bordeaux Sud Ouest, Institut de Mathématiques de Bordeaux, 351 cours de la Liberation, 33405 Talence Cedex, France. Email address: [email protected]
∗∗Postal address: Department of Mathematical Sciences, University of Liverpool, L69 7ZL, Liverpool, UK. Email address: [email protected]
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Abstract

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The purpose of this paper is to study an optimal stopping problem with constraints for a Markov chain with general state space by using the convex analytic approach. The costs are assumed to be nonnegative. Our model is not assumed to be transient or absorbing and the stopping time does not necessarily have a finite expectation. As a consequence, the occupation measure is not necessarily finite, which poses some difficulties in the analysis of the associated linear program. Under a very weak hypothesis, it is shown that the linear problem admits an optimal solution, guaranteeing the existence of an optimal stopping strategy for the optimal stopping problem with constraints.

Keywords

Optimal stoppingconstrained controlMarkov chainlinear programconvex analytic approachMarkov decision processoccupation measure

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 947 - 966
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research partially supported by the Royal Society, grant no. TG091905.

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