Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Víctor Rivero

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 1081-1102.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098441

Digital Object Identifier
doi:10.1214/12-AIHP477

Mathematical Reviews number (MathSciNet)
MR3052404

Zentralblatt MATH identifier
1266.60086

Subjects
Primary: 60G51(60F99)

Keywords
Convolution equivalent distributions Exponential functionals of Lévy processes Fluctuation theory of Lévy processes

Citation

Rivero, Víctor. Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1081--1102. doi:10.1214/12-AIHP477. https://projecteuclid.org/euclid.aihp/1353098441


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