Bivariate Bernstein-gamma functions and moments of exponential functionals of subordinators

Abstract

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of L\'evy processes. In more detail, for a subordinator (a non-decreasing L\'evy process) $(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t e^{-X_s}ds $, evaluated at a finite, deterministic time $t>0$. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time $t$ which under very minor restrictions is shown to be equivalent to an infinite series. We believe this work can be regarded as a stepping stone towards a more in-depth study of general exponential functionals of L\'evy processes on a finite time horizon.