Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T17:28:34.187Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

[1] Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–87.CrossRefGoogle Scholar
[2] Abate, J., Choudhury, G. L. and Whitt, W. (2000). An introduction to numerical transform inversion and its application to probability models. In Computational Probability, ed. Grassman, W. K., Kluwer, Norwell, NA, pp. 257–323.CrossRefGoogle Scholar
[3] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.CrossRefGoogle Scholar
[4] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[5] Bertoin, J. (1999). Subordinators: examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997; Lecture Notes Math. 1717), Springer, Berlin, pp. 1–91.CrossRefGoogle Scholar
[6] Bulirsch, R. and Stoer, J. (1966). Asymptotic upper and lower bounds for results of extrapolation methods. Numer. Math, 8, 93–104.CrossRefGoogle Scholar
[7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[8] Frolov, G. A. and Kitaev, M. Y. (1998). Improvement of accuracy in numerical methods for inverting Laplace transforms based on the Post–Widder formula. Comput. Math. Appl. 36, 23–34.CrossRefGoogle Scholar
[9] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam.Google Scholar
[10] Jagerman, D. L. (1982). An inversion technique for the Laplace transform. Bell System Tech. J. 61, 1995–1995.CrossRefGoogle Scholar
[11] Joyce, D. C. (1971). Survey of extrapolation processes in numerical analysis. SIAM Rev. 13, 435–490.CrossRefGoogle Scholar
[12] Kovacs, M. and Meerschaert, M. M. (2006). Ultrafast subordinators and their hitting times. Publ. L'Institut Math. 94, 193–206.CrossRefGoogle Scholar
[13] Meerschaert, M. M. and Scheffler, H.-P. (2006). Stochastic model for ultraslow diffusion. Stoch. Process. Appl. 116, 1213–1235.CrossRefGoogle Scholar
[14] Nolan, J. P. (1997). Numerical calculation of stable densities and distribution functions. Commun. Statist. Stoch. Models 13, 759–774.CrossRefGoogle Scholar
[15] Oldham, K., Myland, J. and Spanier, J. (2009). An Atlas of Functions, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[16] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Springer.CrossRefGoogle Scholar
[17] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math 1875). Springer, Berlin.Google Scholar
[18] Riordan, J. (2002). An Introduction to Combinatorial Analysis. Dover Publications, Mineola, NY.Google Scholar
[19] Roman, S. (1984). The Umbral Calculus (Pure Appl. Math. 111). Academic Press, New York.Google Scholar
[20] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
[21] Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar
[22] Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line (Monogr. Textbooks Pure Appl. Math. 259). Marcel Dekker, New York.Google Scholar
[23] Veillette, M. and Taqqu, M. S. (2010). Distribution functions of Poisson random integrals: Analysis and computation. Preprint. Available at http://arxiv.org/abs/1004.5338v1.Google Scholar
[24] Veillette, M. and Taqqu, M. S. (2010). Numerical computation of first-passage times of increasing Lévy processes. Methodology Comput. Appl. Prob. 12, 695–729.CrossRefGoogle Scholar