Blanchet, Jose & Liu, Jingchen. Rare-event Simulation for Multidimensional Regularly Varying Random Walks.
Abstract
We consider the problem of effi cient estimation via simulation of first passage time probabilities for a multidimensional random walk with regularly varying increments. In addition of being a natural gen# eralization of the problem of computing ruin probabilities in insurance nin which the focus is a one dimensional random walk nthis problem captures important features of large deviations for multidimensional heavy# tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state# dependent importance sampling estimator for this class of mul# tidimensional problems. Then, we argue using techniques based on Lyapunov inequalities that our estimator is strongly effi cient in the sense that the mean square error of our estimator can be made ar# bitrarily small by increasing the number of replications, uniformly as the probability of interest approaches zero. An important feature of our algorithm involves the interplay between large deviations for reg# ularly varying processes and linear programming. When the target set is the union of half# spaces, our sampler, which can be described in terms of mixtures, can be shown to approximate in total variation the conditional distribution of the walk given that it hits the target set in finite time.
Authors
Jose Blanchet, Jingchen Liu
Publication date
2009/1
