On certain model of claim arrival

We shall present a novel model for claim arrival in an insurance company. The model is derived from an assumption that there are n independent risks in company’s portfolio and that each risk can produce only one claim. Further, we assume also that the probability of claim’s occurrence during time interval [t; t + _] is the same for all risks (given that claims produced by these risks have not yet occurred) and depends only on _. The derived model differs from (non-homogeneous) Poisson model and can not be encompassed in the Sparre-Andersen model. We will present some features of the model- finite dimensional distributions of the process (Lt), where Lt is the number of claims occurred till the time t, evolution of mean, evolution of variance as well as its covariance and correlation function. We will also discuss a more general model – a pure birth process, which contains the obtained model as a special case. We will present Monte Carlo techniques which are suitable to deal with the more general model. We give explicit (but complicated) formulae for distribution of Lt in the more general model, which may be obtained with the use of inverse Fourier transform techniques and then with the method of residues on complex plane. At the end we will give conditions implying the existence of moments of Lt in the more general model, obtained with the use of Laplace transform techniques.