On the Ruin Problem of Collective Risk Theory

Abstract

The theory of collective risk deals with an insurance business, for which, during a time interval $(0, t)$ (1) the total claim $X(t)$ has a compound Poisson distribution, and (2) the gross risk premium received is $\lambda t$. The risk reserve $Z(t) = u + \lambda t - X(t)$, with the initial value $Z(0) = u$, is a temporally homogeneous Markov process. Starting with the initial value $u$, let $T$ be the first subsequent time at which the risk reserve becomes negative, i.e., the business is "ruined". The problem of ruin in collective risk theory is concerned with the distribution of the random variable $T$; this distribution has not so far been obtained explicitly except in a few particular cases. In this paper, the whole problem is re-examined, and explicit results are obtained in the cases of negative and positive processes. These results are then extended to the case where the total claim $X(t)$ is a general additive process.