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November 2004 Ruin probabilities and overshoots for general Lévy insurance risk processes
Claudia Klüppelberg, Andreas E. Kyprianou, Ross A. Maller
Ann. Appl. Probab. 14(4): 1766-1801 (November 2004). DOI: 10.1214/105051604000000927


We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Lévy processes.


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Claudia Klüppelberg. Andreas E. Kyprianou. Ross A. Maller. "Ruin probabilities and overshoots for general Lévy insurance risk processes." Ann. Appl. Probab. 14 (4) 1766 - 1801, November 2004.


Published: November 2004
First available in Project Euclid: 5 November 2004

zbMATH: 1066.60049
MathSciNet: MR2099651
Digital Object Identifier: 10.1214/105051604000000927

Primary: 60J30 , 60K05 , 60K15 , 90A46
Secondary: 60E07 , 60G17 , 60J15

Keywords: conditional limit theorem , Convolution equivalent distributions , First passage time , heavy tails , insurance risk process , ladder process , Lévy process , overshoot , ruin probability , Subexponential distributions

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.14 • No. 4 • November 2004
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