Strongly Efficient Rare-Event Simulation for Regularly Varying Lévy Processes with Infinite Activities

In this paper, we address rare-event simulation for heavy-tailed Lévy 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 Lévy 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 Lévy processes, the Asmussen–Rosiński approximation, and the debiased multilevel Monte Carlo technique. Through novel characterization for the continuity of the extrema of Lévy processes, we show that the proposed algorithm is unbiased and strongly efficient under mild conditions and hence applicable to a broad class of Lévy processes. In numerical experiments, our algorithm demonstrates significant improvements in efficiency compared with the crude Monte Carlo approach.

Funding: This research was supported by the National Science Foundation [Award CMMI-2146530], and the U.S. Department of Energy, Office of Science [Award Number DE-SC0026326].

Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2024.0627.