On a class of discrete time renewal risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete Km distribution, i.e., the probability generating function (p.g.f.) of the distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. A recursive formula is derived for the expected discounted penalty (Gerber-Shiu) function, which can be used to analyze many quantities associated with the time of ruin, e.g., the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support. Keywords: Sparre Andersen risk process, Km family of distributions, Martingale, Generating function, Generalized Lundberg equation, Recursive formula