Some Comments on the Sparre Andersen Model in the Risk Theory

Abstract

SummaryThe Sparre Andersen model assumes that the interclaim times and the amounts of claims are independent random variables, the former identically distributed according to a distribution function K(t), t ≥ o, K(o) = o, the latter identically distributed according to a distribution function P(y) — ∞ < y < ∞ As is well known, the Poisson risk process corresponds to the particular case K(t) = 1 — eβt. In the present paper it is pointed out that another particular case, viz. K(t) = є(t — h), corresponding to a fixed (and thus — strictly speaking—nonrandom) mterclaim time, h, has interesting applications. Thus, the ruin problem considered by Giezendanner, Straub and Wettenschwiler in a paper to the 1972 International Congress of Actuaries in Oslo can be formulated by means of this particular case. The same can be said about the earlier model brought forward by Ammeter in his 1948 paper in Skandinavisk Aktuarietidskrift.About the contents of the paper the following further information may be given. The general Sparre Andersen model is first presented and then the ruin formulas are given for the case with a positive gross risk premium. Thereafter, a modified and more direct method for deriving certain necessary auxiliary functions is illustrated by examples including 1 a the Giezendanner—Straub—Wettenschwiler model. The rest of the paper contains a discussion from the point of view of the Sparre Andersen model of (1) the discrete (equidistant) inspection of a Poisson process for ruin, (11) the Ammeter model and analogous models, and (111) the Giezendanner—Straub—Wettenschwiler model.