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A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables

Part of: Probability theory and stochastic processes Distribution theory - Probability Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Mark S. Veillette  and
Murad S. Taqqu
Mark S. Veillette*
Affiliation:
Boston University
Murad S. Taqqu*
Affiliation:
Boston University
*
∗Postal address: Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA.
∗Postal address: Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA.
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Abstract

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We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-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 nonnegative Lévy processes.

Keywords

Infinitely divisible distributionPost-Widder formulastable distributionstochastic integration

MSC classification

Secondary: 60E07: Infinitely divisible distributions; stable distributions 65C50: Other computational problems in probability 60-08: Computational methods (not classified at a more specific level) 60-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 217 - 237
Copyright
Copyright © Applied Probability Trust 2011 

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