An Outline of a Generalization — started by E. Sparre Andersen — of the classical Ruin Theory

E. Sparre Andersen [1]) presented to the XVth International Congress of Actuaries, New York, 1957, a model of a collective risk process with a positive gross risk premium where the epochs of claims formed a renewal process. Let Ψ(u) (where u denotes the original risk reserve) denote the ruin probability in this model. Generalizing the classical result Sparre Andersen deduced the inequalitywhere R is a suitable positive number depending on the distribution function (continuous to the right), P(y), — ∞ o, K(o) = o, for the times between the epochs of successive claims, (The times between the epochs of successive claims, the inter-occurrence times, are assumed to be independent and identically distributed random variables. The time between the starting point and the epoch of the first claim is assumed to be independent of and to have the same distribution function as the inter-occurrence times. The amounts of claims are assumed to be independent of each other and of the epochs of claims and to be identically distributed.)