Stationarity aspects of the sparre andersen risk process and the corresponding ruin probabilitles

The Sparre Andersen risk model assumes that the interclaim times (also the time between the origin and the first claim epoch is considered as an interclaim time) and the amounts of claim are independent random variables such that the interclaim times have the common distribution function K(t), t|>/ 0, K(O)= 0 and the amounts of claim have the common distribution function P(y), - ∞ < y < ∞. Although the Sparre Andersen risk process is not a process with strictly stationary increments in continuous time it is asymptotically so if K(t) is not a lattice distribution. That is an immediate consequence of known properties of renewal processes. Another also immediate consequence of such properties is the fact that if we assume that the time between the origin and the first claim epoch has not K(t) but as its distribution function (kb1 denotes the mean of K(t)) then the so modified Sparre Andersen process has stationary increments (this works even if K(t) is a lattice distribution). In the present paper some consequences of the above-mentioned stationarity properties are given for the corresponding ruin probabilities in the case when the gross risk premium is positive.