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