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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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References

Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
Bratley, P., Fox, B.L. & Schrage, L. (1987). A guide to simulation, 2nd ed.New York: Springer-Verlag.CrossRefGoogle Scholar
Cochran, W.G. (1977). Sampling techniques. New York: Wiley.Google Scholar
Fox, B.L. (1990). Computing cumulative reward to absorption and its gradient: Deterministic versus simulation methods. Technical Report, Mathematics Department, University of Colorado at Denver.Google Scholar
Fox, B.L. & Glynn, P.W. (1990). Discrete-time conversion for simulating finite-horizon Markov processes. SIAM Journal on Applied Mathematics 50: 1457–1473.CrossRefGoogle Scholar
Fox, B.L. & Glynn, P.W. Splitting as conditional Monte Carlo. Manuscript in preparation.Google Scholar
Fox, B.L. & Glynn, P.W. Manuscript in preparation.Google Scholar
Glasserman, P. (1990). Discrete-time ‘inversion’ and derivative estimation for Markov chains. Operations Research Letters 9: 305–313.CrossRefGoogle Scholar
Glynn, P.W. (1987). Likelihood ratio gradient estimation: An overview. In Thesen, A., Grant, H. & Kelton, W. David (eds.), Proceedings of the Winter Simulation Conference. San Diego, CA: Society for Computer Simulation, pp. 366–374.CrossRefGoogle Scholar
Goyal, A., Shahabuddin, P., Heidelberger, P., Nicola, V.F. & Glynn, P.W. (1989). Unified framework for simulating Markovian models of highly dependable systems. Technical Report RC 14772, IBM Research Division, Yorktown Heights, NY.Google Scholar
L'Ecuyer, P. (1990). A unified view of the IPA, SF and LR gradient estimation techniques. Management Science 36: 1364–1383.CrossRefGoogle Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, 2nd ed.New York: Wiley.CrossRefGoogle Scholar
Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar