Abstract

This thesis studies ruin probabilities and ruin related quantities, using a unified treatment of analysis through the celebrated Gerber-Shiu (G-S) penalty function. For different insurance risk models, a G-S function discounts a penalty due at ruin, which may depend on the surplus before ruin and the deficit at ruin. These insurance risk models include Sparre Andersen's risk model, both in a continuous and in a discrete time setting, diffusion perturbed Sparre Andersen models, as well as risk models with a constant dividend barrier. All these models are extensions of the classical risk model and of diffusion perturbed classical risk model. These G-S penalty functions, considered as functions of initial surplus, satisfy certain integral equations or integro-differential equations, which can be solved to yield defective renewal equations. Such defective renewal equations have a natural probabilistic interpretation, which relies heavily on the roots to a generalized Lundberg's fundamental equation that have a positive real part. These generalized Lundberg equations are from an appropriately chosen exponential martingale. The defective renewal equations (also called recursive formulas in discrete models), that the expected penalty functions satisfy, allow the use of the existing techniques in renewal theory. They can be used to analyze many quantities associated with the time of ruin, such as explicit expressions, bounds, approximations and asymptotic formulas for ruin probabilities, the Laplace transform (or gene-rating function in discrete models) of the time of ruin, the discounted joint and marginal distribution of the surplus immediately before ruin and the deficit at ruin, as well as their moments. Finally, explicit results for the G-S discounted penalty function can be solved when the initial reserve is zero and when the claim sizes are rationally distributed, i.e., the Laplace transform of the claim size density is a rational function.