Distributions of the Surplus Before Ruin: the Deficit at Ruin and the Claim Causing Ruin in a Class of Discrete Time Risk Models

We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete Km distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of 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, Compound geometric distribution, Gerber-Shiu function, Surplus before ruin, Deficit at ruin, Claim causing ruin