Abstract

Asymptotics of densities of exponential functionals of subordinators Abstract: In this talk we are going to present  a new class of special functions that we call Bernstein-Gamma functions. Being a natural extension of the classical Gamma function these functions play a role in the study of some Markov self-similar processes and for these reason they have also appeared in fractional calculus as an extension of Caputo’s derivative. We shall make an overview of the current understanding of the properties of Bernstein-Gamma functions.  We are going to employ these  functions for the study the large asymptotic of densities of exponential functionals of subordinators. The intense study of exponential functionals of L ́evy processes has been triggered by the different applications these quantities have both in theoretical and applied studies. Gradually, their large asymptotic has been almost completely understood with the omission of the case when the underlying L ́evy processes is a subordinator. This is due to the fact that the asymptotic is non-classical from Tauberian point of view. The method we employ is based on the saddle point method which involves the fine understanding of the Stirling-type asymptotic of  the  Bernstein-Gamma functions. This is joint work with Martin Minchev