Open Access
2016 A note on a Poissonian functional and a $q$-deformed Dufresne identity
Reda Chhaibi
Electron. Commun. Probab. 21: 1-13 (2016). DOI: 10.1214/16-ECP4055


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.


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Reda Chhaibi. "A note on a Poissonian functional and a $q$-deformed Dufresne identity." Electron. Commun. Probab. 21 1 - 13, 2016.


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

zbMATH: 1339.33018
MathSciNet: MR3492930
Digital Object Identifier: 10.1214/16-ECP4055

Primary: 33D05 , 60J27 , 60J65

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

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