Strongly Efficient Rare-Event Simulation for Regularly Varying Lévy Processes with Infinite Activities
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917741621936128 |
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| author | Wang, Xingyu Rhee, Chang-Han |
| author_facet | Wang, Xingyu Rhee, Chang-Han |
| contents | 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 randomized debiasing technique. By establishing a novel characterization for the Lipschitz continuity of the law 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 to the crude Monte-Carlo approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_13820 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strongly Efficient Rare-Event Simulation for Regularly Varying Lévy Processes with Infinite Activities Wang, Xingyu Rhee, Chang-Han Probability 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 randomized debiasing technique. By establishing a novel characterization for the Lipschitz continuity of the law 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 to the crude Monte-Carlo approach. |
| title | Strongly Efficient Rare-Event Simulation for Regularly Varying Lévy Processes with Infinite Activities |
| topic | Probability |
| url | https://arxiv.org/abs/2309.13820 |