Some Aspects of Cumulative Risk

The classical treatment of stochastic models in non-life insurance is to first derive the well-known Poisson distribution by considering the question of how many claims take place during a definite period t. In deriving this distribution the following three assumptions are made:( I) The population studied is homogeneous.( II) The occurrence of a claim is a rare event, viz. in an infinite-simal time interval [t, t + Δt], the probability of more than one occurrence must be of the order of magnitude o(Δt).(III) The occurrence of any later claim is not influenced by previous ones (no contagion).In my purely theoretical study [5]), the consequences of discarding one or more of the above assumptions were considered. By so generalizing the Poisson distribution, a great many stochastic models can be built, although the results were not always successful. The following study concentrates on some considerations based on assumption (II). The theoretical formulation of the model will be dealt with briefly and the author would first make reference to the instructive article of Thyrion [7] which was unfortunately unknown to him when he was preparing his already mentioned paper. Ammeter [2] and Arfwedson [3] have also considered special cases of this generalization. With the help of the statistics over traffic accidents in the city of Zurich, I hope to throw some more light on the practical aspects of the problem. To the Statistics Office of this city I would express my thanks for kindly placing all documents at my disposal.