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Overshoots and undershoots of Lévy processes. (English) Zbl 1101.60029
The authors deal with fluctuation problems of a Lévy process defined on the filtered space \((\Omega \text{ Fourier} ,F,P)\) where the filtration satisfies the assumptions of right continuity and completion.
Authors’ 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 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 C. Klüppelberg, A. E. Kyprianou and R. A. Maller [Ann. Appl. Probab. 14, 1766–1801 (2004; Zbl 1066.60049)] 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 side.

MSC:
60G51 Processes with independent increments; Lévy processes
60G50 Sums of independent random variables; random walks
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