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.
Publié le : 2004-11-14
Classification:
Insurance risk process,
Lévy process,
conditional limit theorem,
first passage time,
overshoot,
ladder process,
ruin probability,
subexponential distributions,
convolution equivalent distributions,
heavy tails,
60J30,
60K05,
60K15,
90A46,
60E07,
60G17,
60J15
@article{1099674077,
author = {Kl\"uppelberg, Claudia and Kyprianou, Andreas E. and Maller, Ross A.},
title = {Ruin probabilities and overshoots for general L\'evy insurance risk processes},
journal = {Ann. Appl. Probab.},
volume = {14},
number = {1},
year = {2004},
pages = { 1766-1801},
language = {en},
url = {http://dml.mathdoc.fr/item/1099674077}
}
Klüppelberg, Claudia; Kyprianou, Andreas E.; Maller, Ross A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab., Tome 14 (2004) no. 1, pp. 1766-1801. http://gdmltest.u-ga.fr/item/1099674077/