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Stochastic monotonicity and conditional Monte Carlo for likelihood ratios

Part of:Distribution theory - ProbabilityMarkov processesMultivariate analysisProbabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Paul Glasserman
Paul Glasserman*
Affiliation:
Columbia University
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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 orderings60J10: Markov chains (discrete-time Markov processes on discrete state spaces)62H20: Measures of association (correlation, canonical correlation, etc.)

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993, pp. 103 - 115
Copyright
Copyright © Applied Probability Trust 1993 

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