Distribution Functions of Poisson Random Integrals: Analysis and Computation

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral I(g) = ò₀^T g(s) N(ds)I(g) = displaystyle int_0^T g(s) N(ds), where N is a Poisson random measure with control measure n and g is a suitable kernel function. We do so by combining a Kolmogorov–Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.