Measure-Valued Differentiation for Stationary Markov Chains

Authors:
Bernd Heidergott;Arie Hordijk;Heinz Weisshaupt
Affiliations:
Vrije Universiteit and Tinbergen Institute, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands;Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RA Leiden, The Netherlands;Department of Statistics, University of Vienna, Universitaetsstrasse 5/3, A-1010 Vienna, Austria
Venue:
Mathematics of Operations Research
Year:
2006

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Abstract

We study general state-space Markov chains that depend on a parameter, say, . Sufficient conditions are established for the stationary performance of such a Markov chain to be differentiable with respect to . Specifically, we study the case of unbounded performance functions and thereby extend the result on weak differentiability of stationary distributions of Markov chains to unbounded mappings. First, a closed-form formula for the derivative of the stationary performance of a general state-space Markov chain is given using an operator-theoretic approach. In a second step, we translate the derivative formula into unbiased gradient estimators. Specifically, we establish phantom-type estimators and score function estimators. We illustrate our results with examples from queueing theory.