The ruin problem for mixed poisson risk processes

Abstract

Abstract In the classical model of collective risk theory the individual claims occur in a simple Poisson process. This paper investigates the ruin problem under the more general hypothesis that the claims occur in a mixed Poisson process. Details are given of the numerical calculation of the probability of ruin. In addition an upper bound for this probability is conjectured. The inequality is illustrated by some numerical calculations. To avoid misunderstandings I should mention that the terminology used is that of McFadden (1965) and Biihlmann (1968). Hence a “mixed Poisson” random variable is a Poisson random variable whose mean is itself a random variable, whereas a “compound Poisson” random variable is a random sum of identically distributed random variables, the number of summands being a Poisson variable.