Distribution functions of Poisson random integrals: Analysis and computation

TL;DR

This paper develops a numerical method combining a Kolmogorov-Feller equation and finite differences to compute the distribution of Poisson stochastic integrals, providing convergence rates and practical examples.

Contribution

It introduces a novel numerical approach for calculating the distribution functions of Poisson integrals, with proven convergence and implementation details.

Findings

Convergence rate of the numerical scheme is established.

Method successfully applied to various example integrals.

Software implementation demonstrated in the paper.

Abstract

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral $I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds)$, where $N$ is a Poisson random measure with control measure $n$ and $\krnl$ 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.