Overshoots and undershoots of Lévy Processes

Doney, R. A. and Kyprianou, A. E. (2006) Overshoots and undershoots of Lévy Processes. [MIMS Preprint]

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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 et al. (2004) 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:	MIMS Preprint
Uncontrolled Keywords:	Levy 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:	Dr Mark Muldoon
Date Deposited:	13 Apr 2006
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/221

Available Versions of this Item

• Overshoots and undershoots of Lévy Processes. (deposited 13 Apr 2006) [Currently Displayed]
  • Overshoots and undershoots of Lèvy processes. (deposited 18 Oct 2006)

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