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On extreme ruinous behaviour of Lévy insurance risk processes

Part of: Markov processes Special processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

C. Klüppelberg  and
A. E. Kyprianou
C. Klüppelberg*
Affiliation:
Munich University of Technology
A. E. Kyprianou*
Affiliation:
Heriot-Watt University
*
∗Postal address: Center for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, D-85747 Garching, Germany. Email address: cklu@ma.tum.de
∗∗Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: kyprianou@ma.hw.ac.uk
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Abstract

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In this short note we show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Klüppelberg et al. (2004) and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distribitions for a general class of Lévy insurance risk processes. The results bring the earlier conclusions of Asmussen and Klüppelberg (1996) for the Cramér-Lundberg process into greater generality.

Keywords

Lévy processinsurance risk processruinextreme value theory

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 91B30: Risk theory, insurance
Secondary: 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 594 - 598
Copyright
© Applied Probability Trust 2006 

References

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