A Saddlepoint Approximation to the Distribution of Inhomogeneous Discounted Compound Poisson Processes

In this article we propose an accurate approximation to the distribution of the discounted total claim amount, 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. More precisely, we compute cumulant generating functions of such discounted total claim amounts under various intensity functions and individual claim amount distributions, and invert them by the saddlepoint approximation. We provide precise conditions under which the saddlepoint approximation holds. The resulting approximation is numerically accurate, computationally fast and hence more efficient than Monte Carlo simulation.