## Electronic Communications in Probability

### A note on a Poissonian functional and a $q$-deformed Dufresne identity

Reda Chhaibi

#### Abstract

In this note, we compute the Mellin transform of a Poissonian exponential functional, the underlying process being a simple continuous time random walk. It shows that the Poissonian functional can be expressed in term of the inverse of a $q$-gamma random variable.

The result interpolates between two known results. When the random walk has only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor. In the Brownian limit ($q \rightarrow 1^-$), one recovers Dufresne’s identity involving an inverse gamma random variable. Hence, one can see it as a $q$-deformed Dufresne identity.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 35, 13 pp.

Dates
Accepted: 11 April 2016
First available in Project Euclid: 21 April 2016

https://projecteuclid.org/euclid.ecp/1461251163

Digital Object Identifier
doi:10.1214/16-ECP4055

Mathematical Reviews number (MathSciNet)
MR3492930

Zentralblatt MATH identifier
1339.33018

#### Citation

Chhaibi, Reda. A note on a Poissonian functional and a $q$-deformed Dufresne identity. Electron. Commun. Probab. 21 (2016), paper no. 35, 13 pp. doi:10.1214/16-ECP4055. https://projecteuclid.org/euclid.ecp/1461251163

#### References

• [1] Larbi Alili, Wissem Jedidi, and Victor Rivero, On exponential functionals, harmonic potential measures and undershoots of subordinators, ALEA: Latin American Journal of Probability and Mathematical Statistics 11 (2014), no. 2, 711–735.
• [2] Anita Behme and Alexander Lindner, On exponential functionals of lévy processes, Journal of Theoretical Probability (2013), 1–40 (English).
• [3] Christian Berg, On a generalized gamma convolution related to the $q$-calculus, Theory and applications of special functions, Dev. Math., vol. 13, Springer, New York, 2005, pp. 61–76.
• [4] Jean Bertoin, Philippe Biane, and Marc Yor, Poissonian exponential functionals, $q$-series, $q$-integrals, and the moment problem for log-normal distributions, Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. Probab., vol. 58, Birkhäuser, Basel, 2004, pp. 45–56.
• [5] Jean Bertoin, Alexander Lindner, and Ross Maller, On continuity properties of the law of integrals of Lévy processes, Séminaire de probabilités XLI, Lecture Notes in Math., vol. 1934, Springer, Berlin, 2008, pp. 137–159.
• [6] Jean Bertoin and Marc Yor, Exponential functionals of Lévy processes, Probab. Surv. 2 (2005), 191–212.
• [7] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.
• [8] Ralph Philip Boas, Jr., Entire functions, Academic Press Inc., New York, 1954.
• [9] Philippe Carmona, Frédérique Petit, and Marc Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, Exponential functionals and principal values related to Brownian motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997, pp. 73–130.
• [10] Reda Chhaibi, Beta-Gamma algebra identities and Lie-theoretic exponential functionals of Brownian motion, Electron. J. Probab. 20 (2015), no. 108, 1–20.
• [11] Alberto De Sole and Victor G. Kac, On integral representations of $q$-gamma and $q$-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 1, 11–29.
• [12] Daniel Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. (1990), no. 1-2, 39–79.
• [13] Eberhard Freitag and Rolf Busam, Complex analysis, second ed., Universitext, Springer-Verlag, Berlin, 2009.
• [14] George Gasper and Mizan Rahman, Basic hypergeometric series, second ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004, With a foreword by Richard Askey.
• [15] Francis Hirsch and Marc Yor, On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator, Bernoulli 19 (2013), no. 4, 1350–1377.
• [16] Harry Kesten, Random difference equations and renewal theory for products of random matrices, Acta Mathematica 131, no. 1, 207–248.
• [17] Andreas E. Kyprianou, Fluctuations of Lévy processes with applications, second ed., Universitext, Springer, Heidelberg, 2014, Introductory lectures.
• [18] Alexander Lindner and Ken-iti Sato, Properties of stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes, Math. Nachr. 284 (2011), no. 17–18, 2225–2248.
• [19] Anthony G. Pakes, Length biasing and laws equivalent to the log-normal, J. Math. Anal. Appl. 197 (1996), no. 3, 825–854.
• [20] J. C. Pardo, P. Patie, and M. Savov, A Wiener-Hopf type factorization for the exponential functional of Lévy processes, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 930–956.
• [21] Pierre Patie and Mladen Savov, Extended factorizations of exponential functionals of Lévy processes, Electron. J. Probab. 17 (2012), no. 38, 22.
• [22] Pierre Patie and Mladen Savov, Exponential functional of Lévy processes: generalized Weierstrass products and Wiener-Hopf factorization, C. R. Math. Acad. Sci. Paris 351 (2013), no. 9–10, 393–396.