Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case

We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Levy measure of the underlying Levy process. The method of proof relies on fluctuation theory of Levy processes and an explicit path-wise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish rather general estimates of the measure of the excursions out from zero for the underlying Levy process reflected in its past infimum, whose area under the exponential of the excursion path exceed a given value.