Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T06:35:52.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic monotonicity and conditional Monte Carlo for likelihood ratios

Part of: Markov processes Distribution theory - Probability Probabilistic methods, simulation and stochastic differential equations Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Paul Glasserman
Paul Glasserman*
Affiliation:
Columbia University
Get access Rights & Permissions [Opens in a new window]

Abstract

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias.

We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.

Keywords

VARIANCE REDUCTIONIMPORTANCE SAMPLINGMONOTONE LIKELIHOOD RATIOASSOCIATED RANDOM VARIABLESSTOCHASTICALLY MONOTONE MARKOV CHAINS

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60E15: Inequalities; stochastic orderings 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 103 - 115
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Asmussen, S. and Rubinstein, R. Y. (1992) The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models. Adv. Appl. Prob. 24, 172–201.Google Scholar
[2] Brémaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
[3] Brown, M., Solomon, H. and Stephens, M. A. (1981) Monte Carlo simulation of the renewal function. J. Appl. Prob. 18, 426–434.Google Scholar
[4] Daley, D. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.Google Scholar
[5] Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables. Ann. Math. Statist. 38, 1466–1474.Google Scholar
[6] Glasserman, P. (1992) Processes with associated increments. J. Appl. Prob. 29, 313–333.Google Scholar
[7] Glynn, P. W. and Iglehart, D. L. (1988) Simulation methods for queues: an overview. QUESTA 3, 221–256.Google Scholar
[8] Glynn, P. W. and Iglehart, D. L. (1989) Importance sampling for stochastic simulations. Management Sci. 35, 1367–1392.Google Scholar
[9] Harris, T. E. (1977) A correlation inequality for Markov processes on partially ordered spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
[10] Heidelberger, G. and Iglehart, D. L. (1979) Comparing stochastic systems using regenerative simulation and common random numbers. Adv. Appl. Prob. 11, 804–819.CrossRefGoogle Scholar
[11] Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[12] Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities: I. multivariate totally positive distributions. J. Multivarate Anal. 10, 467–498.Google Scholar
[13] Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov chains. Stoch. Proc. Appl. 5, 231–241.CrossRefGoogle Scholar
[14] Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.Google Scholar
[15] Liggett, T. M. (1985) Interacting Partial Systems. Springer-Verlag, New York.Google Scholar
[16] Lindqvist, B. H. (1988) Association of probability measures on partially ordered sets. J. Multivariate Anal. 26, 111–132.CrossRefGoogle Scholar
[17] Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
[18] Massey, W. A. (1987) Stochastic orderings for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.Google Scholar
[19] Ross, S. M. (1988) Simulating average delay—variance reduction by conditioning, Prob. Eng. Inf. Sci. 2, 309–312.Google Scholar
[20] Shaked, M. and Shanthikumar, J. G. (1988) Temporal stochastic convexity and concavity. Stoch. Proc. Appl. 27, 1–20.Google Scholar
[21] Whitt, W. (1982) Multivariate monotone likelihood ratio and uniform conditional stochastic order. J. Appl. Prob. 19, 695–701.CrossRefGoogle Scholar