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

Abstract

We present a method for computing the PDF and CDF of a non-negative infinitely divisible random variable $X$. Our method uses the Lévy-Khintchine representation of the Laplace transform $\mathbb{E} e^{-λX} = e^{-ϕ(λ)}$, where $ϕ$ 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 Lévy processes. Software to implement this method is available from the authors and we illustrate its use at the end of the paper.