Spectrally negative Lévy processes with applications in risk theory.

*(English)*Zbl 0978.60104Risk processes have been studied for almost a century, starting with the pioneering work of Lundberg (1903). The classical risk model (the compound Poisson process) has been generalized in many ways. In the last decade, compound Poisson processes perturbed by diffusion have also been studied by many authors. However, from the point of view of stochastic processes, many risk processes are special Lévy processes with negative jumps. This type of processes is called a spectrally negative Lévy process. The possibility that some general results from spectrally negative Lévy processes can be directly used to study ruin probabilities and related problems in risk theory is natural. There are only a few papers in risk theory using results from Lévy processes. It is, for examplee, the paper by H. Furrer [Scand. Actuarial J. 1998, No. 1, 59-74 (1998)] which considers the ruin probability of risk processes perturbed by \(\alpha\)-stable processes.

In this paper, the classical risk procees and the gamma process, both perturbed by diffusion, are considered. It is shown that the ruin probabilities and other related problems under many risk processes, including the compound Poisson process and the gamma process, can be handled in a unified and simple way. The first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained.

In this paper, the classical risk procees and the gamma process, both perturbed by diffusion, are considered. It is shown that the ruin probabilities and other related problems under many risk processes, including the compound Poisson process and the gamma process, can be handled in a unified and simple way. The first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained.

Reviewer: Anatoly Swishchuk (Kyïv)

##### MSC:

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |