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Perturbation analysis of inhomogeneous finite Markov chains

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  24 March 2016

Bernd Heidergott ,
Haralambie Leahu ,
Andreas Löpker  and
Georg Pflug
Bernd Heidergott*
Affiliation:
Vrije Universiteit Amsterdam
Haralambie Leahu*
Affiliation:
University of Amsterdam
Andreas Löpker*
Affiliation:
Helmut Schmidt University
Georg Pflug*
Affiliation:
University of Vienna
*
* Postal address: Korteweg-de-Vries Institute for Mathematics, University of Amsterdam, Science Park 107, Postbus 94248, 1090 GE Amsterdam, The Netherlands. Email address: haralambie@gmail.com
** Postal address: Department of Economics and Social Sciences, Helmut Schmidt University, Hamburg, 22008, Germany. Email address: lopker@hsu-hh.de
*** Postal address: Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern Platz 1, Vienna, 1090, Austria. Email address: georg.pflug@univie.ac.at
**** Postal address: Department of Econometrics and Operations Research and Tinbergen Institute, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Email address: b.f.heidergott@vu.nl
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Abstract

In this paper we provide a perturbation analysis of finite time-inhomogeneous Markov processes. We derive closed-form representations for the derivative of the transition probability at time t, with t > 0. Elaborating on this result, we derive simple gradient estimators for transient performance characteristics either taken at some fixed point in time t, or for the integrated performance over a time interval [0 , t]. Bounds for transient performance sensitivities are presented as well. Eventually, we identify a structural property of the derivative of the generator matrix of a Markov chain that leads to a significant simplification of the estimators.

Keywords

Finite Markov processinhomogeneous Markov processsensitivity analysisinfinitesimal generator

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 65C05: Monte Carlo methods
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 255 - 273
Copyright
Copyright © Applied Probability Trust 2016 

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