Cluster Point Processes

In this chapter we consider a cluster point process model. This means that at every point of a point process on (0, ∞) a cluster of activities starts. We interpret this point as the arrival time of a claim which triggers a random stream of payments from the insurer to the insured. This model includes the chain ladder which is used by many practicing actuaries for forecasting the claim numbers and total claim amounts in future periods, given the total claim amounts and claim numbers from previous years. After the definition of the general cluster model, in Section 11.2 we immediately turn to the chain ladder model. The treatment of chain ladder techniques does not really require genuine point process techniques. It is however convenient to use the language of point processes as a general framework, in order to contrast and compare with the theoretical models considered in Section 11.3. In this section we study processes where the clusters start at the points of a homogeneous Poisson process. A Poisson point represents the arrival time of a claim, whereas the newly started process describes the times and amounts of the payments from the insurer to the insured for this particular claim. Since the resulting claim number and total claim amount processes have Poisson integral structure we can derive their first and second order moments. We also discuss the problem of predicting future claim numbers and total claim amounts given the past number of payments in the special case when the payment process is Poisson.