On the Ruin Problem of Collective Risk Theory
0. Summary. 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 Xt. The risk reserve Z(t) = u + Xt - X(t), with the initial value Z(O) = 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. 1. Introduction. The theory of collective risk, as developed by the Swedish actuary Filip Lundberg, deals with the business of an insurance company. Following a series of papers published by him during the years 1909-1934, a considerable amount of work has been done by Cram6r, Segerdahl, Tacklind, Sax6n, Arfwedson and many others; a survey of the theory from the point of view of stochastic processes was given by Cram6r [2], [3] and an excellent review has recently been given by Arfwedson [1]. Briefly, the mathematical model used in this theory can be described as follows. (a) The claims occur entirely "at random", that is, during the infinitesimal interval of time (t, t + dt), the probability of a claim occurring is dt and the probability of more than one claim occurring is of a smaller order than dt, these probabilities being independent of the claims which have occurred during (0, t). (b) If a claim does occur, the amount claimed is a random variable with the probability distribution dP(x) (-oo < x < m), negative claims occurring in the case of ordinary whole-life annuities. Under the assumptions (a) and (b), it is easily seen that the total amount X(t) of all claims which occur during (0, t) has the compound Poisson distribution given by