Kalman-bucy Filtering for Linear Systems Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Con- Tracts

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the` ltering problem', that is to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy lter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that if the primary event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival and aggregate loss processes by a three-dimensional Gaussian process. We establish weak convergence results. We then use the Kalman-Bucy lter and we obtain the price of reinsurance contracts involving high frequency events.