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Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  19 February 2016

Thorbjörn Gudmundsson  and
Henrik Hult
Thorbjörn Gudmundsson*
Affiliation:
KTH Royal Institute of Technology
Henrik Hult*
Affiliation:
KTH Royal Institute of Technology
*
∗ Postal address: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden.
∗ Postal address: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden.
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Abstract

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In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

Keywords

Markov chain Monte Carloheavy tailrare-event simulationrandom walk

MSC classification

Primary: 65C05: Monte Carlo methods 60J22: Computational methods in Markov chains
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 359 - 376
Copyright
© Applied Probability Trust 

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