Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying L\'evy Processes with Infinite Activities

TL;DR

This paper introduces an efficient importance sampling method for simulating rare events in heavy-tailed L\'evy processes with infinite activities, improving over traditional Monte Carlo techniques.

Contribution

It presents a novel importance sampling algorithm based on large deviations, extrema approximation, and debiasing, applicable to a wide class of L\'evy processes.

Findings

Significant efficiency gains over crude Monte Carlo methods.

Applicable to a broad class of heavy-tailed L\'evy processes.

Validated through numerical experiments.

Abstract

In this paper we address the problem of rare-event simulation for heavy-tailed L\'evy processes with infinite activities. We propose a strongly efficient importance sampling algorithm that builds upon the sample path large deviations for heavy-tailed L\'evy processes, stick-breaking approximation of extrema of L\'evy processes, and the randomized debiasing Monte Carlo scheme. The proposed importance sampling algorithm can be applied to a broad class of L\'evy processes and exhibits significant improvements in efficiency when compared to crude Monte-Carlo method in our numerical experiments.