Technique for computing the PDFs and CDFs of non-negative infinitely divisible random variables

TL;DR

This paper introduces a novel method combining Laplace transform inversion and convergence acceleration to accurately compute PDFs and CDFs of non-negative infinitely divisible random variables, with practical software implementation.

Contribution

The paper presents a new technique leveraging the Lévy-Khintchine representation and Post-Widder method for precise PDF and CDF computation of non-negative infinitely divisible distributions.

Findings

Effective for stable and mixture distributions

Accurate results demonstrated on various examples

Software implementation provided

Abstract

We present a method for computing the PDF and CDF of a non-negative infinitely divisible random variable $X$. Our method uses the L\'{e}vy-Khintchine representation of the Laplace transform $\mathbb{E} e^{-\lambda X} = e^{-\phi(\lambda)}$, where $\phi$ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples including the stable distribution, mixtures thereof, and integrals with respect to non-negative L\'{e}vy processes. Software to implement this method is available from the authors and we illustrate its use at the end of the paper.