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Ruin probabilities and overshoots for general Lévy insurance risk processes. (English) Zbl 1066.60049
From the authors’ abstract: We formulate the insurance risk process in a general Lévy 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 \(-\infty\) 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 S. Asmussen and C. Klüppelberg [Stochastic Processes Appl. 64, 103–125 (1996; Zbl 0879.60020)] and J. Bertoin and R. A. Doney [Adv. Appl. Probab. 28, 207–226 (1996; Zbl 0854.60069)] 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 the further investigation of general renewal-type properties of Lévy processes
Reviewer’s remark: It is worth to note that all authors’ results have exact random walks analogues as well under similar conditions and that they can be proved in a manner similar to that of paper.

60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
60K15 Markov renewal processes, semi-Markov processes
60E07 Infinitely divisible distributions; stable distributions
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