Overshoots and undershoots of Lévy processes

Details Overshoots and undershoots of Lévy processes Overshoots and undershoots of Lévy processes

2006-03-09

arxiv.org/abs/math/0603210v1

Annals of Applied Probability 2006, Vol. 16, No. 1, 91-106

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051605000000647 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051605000000647

We obtain a new fluctuation identity for a general L\'{e}vy 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 Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of L\'{e}vy 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 L\'{e}vy process is spectrally one sided.

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