Simulating heavy tailed processes using delayed hazard rate twisting

Consider the problem of estimating the small probability that the maximum of a random walk exceeds a large threshold, when the process has a negative drift and the underlying random variables may have heavy tailed distributions, that is, their tail distribution decays at a subexponential rate. We consider one class of such problems that has applications in estimating the ruin probability associated with insurance claim processes with subexponentially distributed claim sizes, and in estimating the probability of large delays in an M/G/1 queue with subexponentially distributed service times. Significant work has been done on analogous problems for the light tailed case (when the tail distribution decreases at an exponential rate or faster) that involve importance sampling methods using appropriate exponential twisting. However, for the subexponential case, such exponential twisting is infeasible and alternative techniques are needed. In this paper we introduce importance sampling techniques where the new probability measure is obtained by twisting the hazard rate of the original distribution. For subexponential distributions this amounts to subexponential twisting---twisting at a subexponential rate. In addition, we introduce the technique of "delaying" the twisting and show that the combination of the two techniques produces asymptotically optimal estimates of the small probability mentioned above.