Strongly Efficient Rare-Event Simulation for Regularly Varying L\'evy Processes with Infinite Activities

TL;DR

This paper introduces an efficient rare-event simulation algorithm for heavy-tailed Le9vy processes with infinite activities, overcoming simulation challenges through advanced importance sampling and approximation techniques.

Contribution

It develops a novel, unbiased, and strongly efficient simulation method for complex Le9vy processes with infinite activities, using new theoretical characterizations.

Findings

Significant efficiency improvements over crude Monte Carlo

Algorithm is unbiased and strongly efficient under mild conditions

Applicable to a broad class of heavy-tailed Le9vy processes

Abstract

In this paper, we address rare-event simulation for heavy-tailed L\'evy processes with infinite activities. The presence of infinite activities poses a critical challenge, making it impractical to simulate or store the precise sample path of the L\'evy process. We present a rare-event simulation algorithm that incorporates an importance sampling strategy based on heavy-tailed large deviations, the stick-breaking approximation for the extrema of L\'evy processes, the Asmussen-Rosi\'nski approximation, and the randomized debiasing technique. By establishing a novel characterization for the Lipschitz continuity of the law of L\'evy processes, we show that the proposed algorithm is unbiased and strongly efficient under mild conditions, and hence applicable to a broad class of L\'evy processes. In numerical experiments, our algorithm demonstrates significant improvements in efficiency compared to the crude Monte-Carlo approach.