Overshoots and undershoots of Lèvy processes

Doney, R. A. and Kyprianou, A. E. (2006) Overshoots and undershoots of Lèvy processes. Annals of Applied Probability, 16 (1). pp. 91-106. ISSN 1050-5164

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Abstract

We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided

Item Type: Article
Uncontrolled Keywords: Lévy processes; first passage problem; Wiener–Hopf factorization; insurance risk process
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User: Ms Lucy van Russelt
Date Deposited: 18 Oct 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/612

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