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Path decomposition of a reflected Lévy process on first passage over high levels

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  • Griffin, Philip S.

Abstract

Let X be a real valued Lévy process and set Rt=Xt−infs≤tXs. This paper addresses the asymptotic behavior of the sample paths of the reflected process R on first passage over an arbitrarily high level u. We show that under the convolution equivalent condition of Klüppelberg et al. (2004), the sample paths of R on the first excursion which crosses over a high level u can be decomposed into two processes. The first describes the paths in a neighborhood of the origin. The process then takes a large jump into a neighborhood of u. The second process describes the subsequent paths. This sample path behavior is similar to that of X conditioned to cross level u. Using this connection many results concerning, for example, undershoots and overshoots can be easily obtained.

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  • Griffin, Philip S., 2022. "Path decomposition of a reflected Lévy process on first passage over high levels," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 29-47.
    • Handle: RePEc:eee:spapps:v:145:y:2022:i:c:p:29-47
    DOI: 10.1016/j.spa.2021.11.013
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    References listed on IDEAS

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    1. Griffin, Philip S., 2020. "General tax structures for a Lévy insurance risk process under the Cramér condition," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1368-1387.
    2. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605, October.
    3. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    4. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
    5. Mijatović, Aleksandar & Pistorius, Martijn, 2015. "Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2937-2954.
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