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State-dependent importance sampling for regularly varying random walks

Part of: Algorithms - Computer Science Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jose H. Blanchet  and
Jingchen Liu
Jose H. Blanchet*
Affiliation:
Columbia University
Jingchen Liu*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, S. W. Mudd Building, 500 West 120th Street, New York, NY 10027-6699. Email address: [email protected]
∗∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, Room 1030, New York, NY 10027.
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Abstract

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Consider a sequence (Xk: k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn: n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of Sn in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

Keywords

State dependentimportance samplingregularly varyingrandom walkheavy tailstrong efficiencyrare-event simulation

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60J05: Discrete-time Markov processes on general state spaces 68W40: Analysis of algorithms
Secondary: 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1104 - 1128
Copyright
Copyright © Applied Probability Trust 2008 

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