Poisson Random Measures in Collective Risk Theory

In Chapter 7 we collected the basic notions of point process theory. We have focused on Poisson random measures (PRMs) and their properties. In the present chapter we would like to apply the theory developed there, to models from collective risk theory. In particular, we will make considerable use of the marking and transformation techniques of PRMs introduced in Section 7.3, and we will intensively exploit the independence of Poisson claim numbers and Poisson integrals on disjoint parts of the time-claim size space. In Section 8.1, we consider different decompositions of the time-claim size space, such as decomposition by claim size, year of occurrence, year of reporting, etc. In Section 8.2, we study a major generalization of the Cramér-Lundberg model, called the basic model, which accounts for delays in reporting, claim settlements, as well as the payment process in the settlement period of the claim. We also decompose the time-claim size space into its basic ingredients, resulting in settled, incurred but not reported, and reported but not settled claims. We study the distributions of the corresponding claim numbers and total claim amounts.

This chapter was inspired by the ideas in Norberg’s [114] article on point process techniques for non-life insurance.