Dr. Riccardo Gatto (University of Bern, visiting Pstat)

Title: The Distribution of the Inhomogeneous Discounted Compound Poisson Process

Abstract: We present a practical and accurate approximation to the distribution of the discounted total claim amount generated by a portfolio of risks, where the individual claim amounts are independent and identically distributed and the number of claims over a specified period is governed by an inhomogeneous Poisson process.

We show how to obtain the cumulant generating function of this discounted total claim amount, under various choices for the intensity functions and for individual claim amount distributions. We suggest inverting the cumulant generating function by the saddlepoint approximation of asymptotic analysis. We provide precise conditions under which this saddlepoint approximation exists.

Comparisons with Monte Carlo methods for simulating inhomogeneous Poisson precesses show that the suggested approximation is numerically accurate and computationally fast. An example where a Monte Carlo method breaks down is also given.

The method proposed can be directly extended to more general inhomogeneous shot-noise processes and therefore it has potential applications in various physical sciences.