Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The excursion measure away from zero for spectrally negative Lévy processes

J. C. Pardo, J. L. Pérez, and V. M. Rivero

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We provide a description of the excursion measure from a point for a spectrally negative Lévy process. The description is based in two main ingredients. The first is building a spectrally negative Lévy process conditioned to avoid zero and the study of its entrance law at zero. The latter is connected with both the excursion measure from zero of the process reflected in its infimum and reflected in its supremum. This leads us to establish a connection between the excursion measure from the state zero and the excursion measure from zero for the process reflected at the infimum and reflected at the supremum, respectively, which is the second main ingredient of our description.


Dans ce travail on décrit la mesure d’excursions en dehors de zéro pour les processus de Lévy spectralement négatifs. Cette description utilise deux ingrédients principaux. Le premier consiste à construire un processus de Lévy spectralement négatif conditionné à éviter zéro et étudier sa loi d’entrée. Celle ci est intimement liée aux mesures d’excursions en dehors de zéro du processus réfléchi dans l’infimum passé, et, respectivement, dans le supremum passé. Ceci établit un lien naturel entre la mesure d’excursions en dehors de zéro et ces deux dernières mesures d’excursions. Ceci est le deuxième ingrédient de notre description.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 75-99.

Received: 19 July 2015
Revised: 16 September 2016
Accepted: 19 September 2016
First available in Project Euclid: 19 February 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G17: Sample path properties

Lévy processes Excursion theory from a point Local times Fluctuation theory


Pardo, J. C.; Pérez, J. L.; Rivero, V. M. The excursion measure away from zero for spectrally negative Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 75--99. doi:10.1214/16-AIHP795.

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