Prediction in a Poisson cluster model with multiple cluster processes

We consider a simple but flexible extension of the Poisson cluster model studied in Matsui & Mikosch (2010). In the former, model only a single cluster process starts at each jump point of the Poisson process, whereas we start a randomly given number of cluster processes at each jump. This simple extension yields additional mathematical problems in prediction of future increments of the process which are based on the past observations. However, by making full use of the Poisson structure of the model, we derive reasonably explicit expressions for predictors, which is of critical importance in the insurance application. Some comparisons of predictors are also made by their mean-squared errors when the cluster process is a compound Poisson process. The result yields a natural conclusion that the finer information we use, the better predictors we obtain.