Spectrally negative Lévy processes with applications in risk theory

In this paper, results on spectrally negative Lévy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, 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. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Lévy process theory's point of view and in a unified and simple way.