Asymptotics for densities of exponential functionals of subordinators

Abstract

In this paper we derive non-classical Tauberian asymptotics at infinity for the tail, the density and its derivatives of a large class of exponential functionals of subordinators. More precisely, we consider the case for which the Lévy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein–gamma functions. To apply the saddle point method, we improved the Stirling type asymptotics for Bernstein–gamma functions in the complex plane. As an application we derive the asymptotics of the density and its derivatives for all exponential functionals of a broad class of non-decreasing, potentially killed compound Poisson processes, which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.