On the probability of ruin in the presence of a linear dividend barrier

Abstract

1. Introduction and Summary

We shall consider the model that was introduced in Gerber (1974). Let {S_t } denote the compound Poisson process of the aggregate claims (given by the Poisson parameter λ and the distribution of individual claim amounts F(y), y⩾0), and let c denote the premium density. Of course it is assumed that c>λ , and that the adjustment coefficient R exists. The classical model is now modified as follows: Whenever the surplus reaches a certain barrier, dividends are paid out such that the surplus stays on the barrier (until the next claim occurs). We consider the case where the dividend barrier is a linear function of time, b + at, where b⩾0, 0<a<c. Thus, if Rt denotes the surplus at time t, Together with the knowledge of the initial surplus, R ₀=x(0⩽x⩽b<∞), this determines the process {R_t }.