Author:
Blanchet Jose H.,Liu Jingchen
Abstract
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 ofSnin a large deviation regime asn↗ ∞. 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Statistics and Probability
Cited by
24 articles.
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