Precise asymptotics for the density and the upper tail of exponential functionals of subordinators

TL;DR

This paper derives precise large-time asymptotics for the density and upper tail of exponential functionals of subordinators, enhancing understanding of their probabilistic behavior and applications.

Contribution

It provides exact asymptotic formulas for the density and tail distributions, improving upon previous logarithmic estimates and analyzing the influence of the Lévy measure.

Findings

Exact large-time asymptotics for density and tail distributions

Influence of Lévy measure behavior near zero on asymptotics

Application examples illustrating theoretical results

Abstract

We provide exact large-time equivalents of the density and upper tail distributions of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of the upper tail. Since the results in their general form remain rather abstract, we study some fairly common situations in details to see how the behavior near 0 of the L\'evy measure of the subordinator influences the asymptotics of the density and upper tail distributions. Several examples of applications are then discussed.