Electronic Communications in Probability

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

Reda Chhaibi

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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

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

Received: 16 January 2015
Accepted: 11 April 2016
First available in Project Euclid: 21 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33D05: $q$-gamma functions, $q$-beta functions and integrals 60J27: Continuous-time Markov processes on discrete state spaces 60J65: Brownian motion [See also 58J65]

$q$-calculus $q$-gamma random variable exponential functionals of compound Poisson process $q$-analogue of Dufresne’s identity for exponential functionals of Brownian motion Wiener-Hopf factorization

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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

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