The Annals of Applied Probability

Dynamic importance sampling for uniformly recurrent Markov chains

Paul Dupuis and Hui Wang

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Abstract

Importance sampling is a variance reduction technique for efficient estimation of rare-event probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper we consider dynamic importance sampling in the setting of uniformly recurrent Markov chains. By “dynamic” we mean that in the course of a single simulation, the change of measure can depend on the outcome of the simulation up till that time. Based on a control-theoretic approach to large deviations, the existence of asymptotically optimal dynamic schemes is demonstrated in great generality. The implementation of the dynamic schemes is carried out with the help of a limiting Bellman equation. Numerical examples are presented to contrast the dynamic and standard schemes.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 1A (2005), 1-38.

Dates
First available in Project Euclid: 28 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922319

Digital Object Identifier
doi:10.1214/105051604000001016

Mathematical Reviews number (MathSciNet)
MR2115034

Zentralblatt MATH identifier
1068.60036

Subjects
Primary: 60F10: Large deviations 65C05: Monte Carlo methods 93E20: Optimal stochastic control

Keywords
Asymptotic optimality importance sampling Markov chain Monte Carlo simulation rare events stochastic game weak convergence

Citation

Dupuis, Paul; Wang, Hui. Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 (2005), no. 1A, 1--38. doi:10.1214/105051604000001016. https://projecteuclid.org/euclid.aoap/1106922319


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