Probability Theory
Exponential functional of Lévy processes: Generalized Weierstrass products and Wiener–Hopf factorization
Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 393-396.

In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalized Weierstrass products. As by-product, we obtain a multiplicative Wiener–Hopf factorization generalizing previous results obtained by Patie and Savov (2012) [14] as well as smoothness properties of its distribution.

Dans cette note, nous énonçons une représentation de la transformée de Mellin de la fonctionnelle exponentielle des processus de Lévy sous la forme de produits de Weierstrass généralisés. Nous en déduisons une factorisation multiplicative de Wiener–Hopf généralisant un résultat obtenu récemment par Patie et Savov (2012) [14] ainsi que des propriétés de régularité pour sa loi.

Accepted:
Published online:
DOI: 10.1016/j.crma.2013.04.023

Pierre Patie 1; Mladen Savov 2

1 School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA
2 University of Reading, Department of Mathematics and Statistics, Whiteknights, PO Box 220, Reading RG6 6AX, UK
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Pierre Patie; Mladen Savov. Exponential functional of Lévy processes: Generalized Weierstrass products and Wiener–Hopf factorization. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 393-396. doi : 10.1016/j.crma.2013.04.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.023/

[1] J. Bertoin Lévy Processes, Cambridge University Press, 1996

[2] J. Bertoin; M. Yor On subordinators, self-similar Markov processes and some factorizations of the exponential variable, Electron. Commun. Probab., Volume 6 (2001), pp. 95-106 (electronic)

[3] J. Bertoin; M. Yor Exponential functionals of Lévy processes, Probab. Surv., Volume 2 (2005), pp. 191-212

[4] J. Bertoin; A. Lindner; R. Maller On continuity properties of the law of integrals of Lévy processes, Séminaire de probabilités XLI, Lect. Notes Math., vol. 1934, Springer, 2008, pp. 137-159

[5] Ph. Carmona; F. Petit; M. Yor Beta-gamma random variables and intertwining relations between certain Markov processes, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 311-368

[6] B. Haas; V. Rivero Quasi-stationary distributions and Yaglom limits of self-similar Markov processes, Stoch. Process. Appl., Volume 122 (2012) no. 12, pp. 4054-4095

[7] F. Hirsch, M. Yor, On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator, Preprint, Université dʼEvry, 2011.

[8] A. Kuznetsov; J.C. Pardo Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes, Acta Appl. Math., Volume 123 (2013) no. 1, pp. 113-139 | DOI

[9] N.N. Lebedev Special Functions and Their Applications, Dover Publications, New York, 1972

[10] K. Maulik; B. Zwart Tail asymptotics for exponential functionals of Lévy processes, Stoch. Process. Appl., Volume 116 (2006), pp. 156-177

[11] J.C. Pardo; P. Patie; M. Savov A Wiener–Hopf type factorization for the exponential functional of Lévy processes, J. Lond. Math. Soc., Volume 86 (2012) no. 3, pp. 930-956

[12] P. Patie Law of the absorption time of some positive self-similar Markov processes, Ann. Probab., Volume 40 (2012) no. 2, pp. 765-787

[13] P. Patie, M. Savov, Spectral theory for positive invariant Lamperti-Feller semigroups: The discrete spectrum case via intertwining, Working paper, Available upon request, 2012.

[14] P. Patie; M. Savov Extended factorizations of exponential functionals of Lévy processes, Electron. J. Probab., Volume 17 (2012) no. 38 (22 pp)

[15] K. Sato Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999

[16] M. Yor Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, Berlin, 2001

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