Efficient rare event simulation for heavy-tailed compound sums

Abstract

We develop an efficient importance sampling algorithm for estimating the tail distribution of heavy-tailed compound sums, that is, random variables of the form S M = Z 1 +…+ Z M where the Z i 's are independently and identically distributed (i.i.d.) random variables in R and M is a nonnegative, integer-valued random variable independent of the Z i 's. We construct the first estimator that can be rigorously shown to be strongly efficient only under the assumption that the Z i 's are subexponential and M is light-tailed. Our estimator is based on state-dependent importance sampling and we use Lyapunov-type inequalities to control its second moment. The performance of our estimator is empirically illustrated in various instances involving popular heavy-tailed models.