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

Rivero, Víctor

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1081-1102.

On s'intéresse à la vitesse de décroissance de la queue de distribution d'une fonctionnelle exponentielle d'un processus de Lévy dont la mesure de sauts est équivalente par convolution. Le résultat principal de ce papier montre que cette vitesse décroît comme la queue de la mesure image de la mesure de sauts par la fonction exponentielle. La preuve de ce résultat repose sur la théorie des fluctuations pour les processus de Lévy et une représentation trajectorielle explicite de la fonctionnelle exponentielle comme la fonctionnelle exponentielle d'un subordinateur bivarié. Nos techniques nous permettent également d'établir des résultats similaires sous la mesure d'excursion du processus de Lévy sous-jacent réfléchi en son minimum passé.

We determine the rate of decrease of the right tail distribution of the exponential functional of a Lévy process with a convolution equivalent Lévy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Lévy measure of the underlying Lévy process. The method of proof relies on fluctuation theory of Lévy processes and an explicit pathwise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish analogous results under the excursion measure of the underlying Lévy process reflected in its past infimum.

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