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Estimating Derivatives Via Poisson's Equation

Published online by Cambridge University Press:  27 July 2009

Bennett L. Fox  and
Paul Glasserman
Bennett L. Fox
Affiliation:
Department of MathematicsUniversity of Colorado Denver, Colorado 80217-3364
Paul Glasserman
Affiliation:
Graduate School of Business Columbia University New York, New York 10027
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Abstract

Let x(j) be the expected reward accumulated up to hitting an absorbing set in a Markov chain, starting from state j. Suppose the transition probabilities and the one-step reward function depend on a parameter, and denote by y(j) the derivative of x(j) with respect to that parameter. We estimate y(0) starting from the respective Poisson equations that x = [x(0),x(l),…] and y = [y(0),y(l),…] satisfy. Relative to a likelihood-ratio-method (LRM) estimator, our estimator generally has (much) smaller variance; in a certain sense, it is a conditional expectation of that estimator given x. Unlike LRM, however, we have to estimate certain components of x. Our method has broader scope than LRM: we can estimate sensitivity to opening arcs.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 4 , October 1991 , pp. 415 - 428
Copyright
Copyright © Cambridge University Press 1991

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