Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

An important question in insurance is how to evaluate the probabilities of (non-) ruin of a company over any given horizon of finite length. This paper aims to present some (not all) useful methods that have been proposed so far for computing, or approximating, these probabilities in the case of discrete claim severities. The starting model is the classical compound Poisson risk model with constant premium and independent and identically distributed claim severities. Two generalized versions of the model are then examined. The former incorporates a non-constant premium function and a non-stationary claim process. The latter takes into account a possible interdependence between the successive claim severities. Special attention will be paid to a recursive computational method that enables us to tackle, in a simple and unified way, the different models under consideration. The approach, still relatively little known, relies on the use of remarkable families of polynomials which are of Appell or generalized Appell (Sheffer) types. The case with dependent claim severities will be revisited accordingly.