Abstract

A generalization to the classical risk model is presented. This generalization includes a Lévy process as the aggregate claims process. The compound Poisson process and the diffusion process are particular cases of this more general model. With this model we attempt to bridge two approaches often used in the literature to generalize the classical model. We investigate applications of pure-jump Lévy processes to risk theory, in particular members of the family of generalized hyperbolic processes. We focus our interest in the normal inverse Gaussian process and in the generalized inverse Gaussian process. Both lead to purely discontinuous risk processes with infinite activity, i.e., these processes have an infinite number of small jumps and occasional larger movements. We also present an approximation to the classical risk model when the claim severities belong to the domain of attraction of an extreme distribution, this allows for all kinds of heavy and medium tailed distributions. The model is based on a Lévy process with an underlying Lévy measure proportional to the generalized Pareto distribution. Most of our results rely on properties that are not only valid for Lévy processes but for the larger class of semimartingales. As an illustration, we also introduce an even more general risk process with independent increments that would endow us with a periodic reserve process that can find applications in reinsurance or in the valuation of catastrophe insurance options. Although, this periodic risk process does not belong to the Lévy family of processes, it does belong to the larger family of processes with independent increments. The main contribution of this thesis takes the form of four independent chapters that illustrate the potential of Lévy modeling in risk theory.