Gerber–Shiu Analysis in the Dependent Sparre Andersen Model

Abstract

A substantial generalization of the classical Poisson model is the dependent Sparre Andersen (or renewal risk) model which allows not only for a more general distribution of the times between claims, but may also impact the size of the claim at the end of the interclaim time interval. Special cases include various copula-based models proposed in the literature to describe various types of insurance scenarios, and the case where the interclaim times and claim sizes are statistically independent (referred to here as the independent Sparre Andersen model) has also been considered in much actuarial research. Various structural properties associated with the joint and marginal densities of the time of ruin, the deficit at ruin, and the surplus immediately prior to ruin in all such models are identified through a combination of conditioning arguments, involving the first claim and the first drop in surplus. Finally, the distribution and moments of the distribution of the deficit are derived.