Ruin probabilities and overshoots for general Lévy insurance risk processes

Abstract

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. . This reprint differs from the original in pagination and typographic detail. 1 2 C. KLÜPPELBERG, A. E. KYPRIANOU AND R. A. MALLER t > 0, are modelled as a compound Poisson process, yielding the risk processGENERAL LÉVY INSURANCE RISK PROCESSES 3 the ruin probability can be shown to decrease exponentially fast, in fact, proportional to e −νu , as u → ∞. This result has been extended to a Lévy process setting by Bertoin and Doney [6]. If such a "Lundberg coefficient" ν does not exist, as is the case for subexponential and other "convolution equivalent" distributions (see Section 3), estimates of the ruin probability have been derived by Embrechts, Goldie and Veraverbeke [15], Embrechts and Veraverbeke [18] and Veraverbeke [35]; see [16], Section 1.4. It is also of prime interest to understand the way ruin happens. This question has been addressed by Asmussen [1] for the Cramér case, and more recently by Asmussen and Klüppelberg [3] for the subexponential case. They describe the sample path behavior of the process along paths leading to ruin via various kinds of conditional limit theorems. As expected, the Cramér case and the non-Cramér case are qualitatively quite different; see, for example, [2] and [16], Section 8.3.Our aim is to investigate the non-Cramér case in a general Lévy process setting, which clearly reveals the roles of the various assumptions. Our Lévy process X will start at 0 and be assumed to drift to −∞ a.s., but otherwise is quite general. Upward movement of X represents "claim payments," and the drift to −∞ reflects the fact that "premium income" should outweigh claims. "Ruin" will then correspond to passage of X above a specified high level, u, say. In this scenario, heavy-tailedness of the positive side of the distribution of upward jumps models the occurrence of large, possibly ruinous, claims, and has previously been studied in connection with the assumption of a finite mean for the process. But in general we do not want to restrict the process in this way. A higher rate of decrease of the process to −∞ is more desirable from the insurer's point of view, while allowing a heavier tail for the positive part is in keeping with the possibility of even more extreme events, which indeed are observed in recent insurance data.This ...

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