A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

Abstract

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn [9] satisfies this property and show that it does so if and only if the tail index α of the right tail lies in the interval (1, 3/2).