The Annals of Applied Probability

Ruin probabilities and overshoots for general Lévy insurance risk processes

Claudia Klüppelberg, Andreas E. Kyprianou, and Ross A. Maller

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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.

Article information

Ann. Appl. Probab., Volume 14, Number 4 (2004), 1766-1801.

First available in Project Euclid: 5 November 2004

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Zentralblatt MATH identifier

Primary: 60J30 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 90A46
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G17: Sample path properties 60J15

Insurance risk process Lévy process conditional limit theorem first passage time overshoot ladder process ruin probability subexponential distributions convolution equivalent distributions heavy tails


Klüppelberg, Claudia; Kyprianou, Andreas E.; Maller, Ross A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004), no. 4, 1766--1801. doi:10.1214/105051604000000927.

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