Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, they studied the joint distribution of the time of ruin, the surplus before ruin and the deficit at ruin [Gerber and Shiu (1997, 1998a, 1998b), Gerber and Landry (1998)]. They work with the classical and the perturbed risk models and hint that their results can be extended to gamma and inverse Gaussian risk processes.
In this paper we work out this extension in the context of a more general risk model. The construction of Dufresne et al. (1991) is based on a non–negative, non–increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. Our aim is to extend their work to a generalized risk model driven by an increasing L´evy process. This first paper presents the results for the case when the aggregate claims process is a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian and generalized inverse Gaussian
processes.