Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the asymptotic behavior of the density of the supremum of Lévy processes

Loïc Chaumont and Jacek Małecki

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Abstract

Let us consider a real valued Lévy process $X$ whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of $X$ before any deterministic time $t$ is absolutely continuous on $(0,\infty)$. We show that its density $f_{t}(x)$ is continuous on $(0,\infty)$ if and only if the potential density $h'$ of the upward ladder height process is continuous on $(0,\infty)$. Then we prove that $f_{t}$ behaves at 0 as $h'$. We also describe the asymptotic behaviour of $f_{t}$, when $t$ tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.

Résumé

Soit $X$ un processus de Lévy réel dont les probabilités de transition sont absolument continues par rapport à la mesure de Lebesgue. Dans ce cas, il est connu que la loi du supremum passé avant un temps déterministe $t$ est elle-même absolument continue sur $(0,\infty)$. En supposant de plus que les densités sont bornées, nous montrons que la densité $f_{t}(x)$ du supremum passé est continue en $x$ sur $(0,\infty)$, si et seulement si la densité potentielle $h'(x)$ du subordinateur des hauteurs d’échelle ascendant est continue sur $(0,\infty)$. Nous montrons alors que $f_{t}$ se comporte en 0 de la même manière que $h'$. Nous donnons également une description du comportement asymptotique de $f_{t}$, lorsque $t$ tend vers l’infini. Enfin nous appliquons ces résultats pour étudier le comportement asymptotique de la densité du méandre des processus de Lévy et de la densité de la loi d’entrée des processus de Lévy conditionnés à rester positifs.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1178-1195.

Dates
Received: 9 November 2013
Revised: 26 February 2015
Accepted: 2 March 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723516

Digital Object Identifier
doi:10.1214/15-AIHP674

Mathematical Reviews number (MathSciNet)
MR3531705

Zentralblatt MATH identifier
1350.60042

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes

Keywords
Density Past supremum Asymptotic behaviour Renewal function Conditioning to stay positive Meander

Citation

Chaumont, Loïc; Małecki, Jacek. On the asymptotic behavior of the density of the supremum of Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1178--1195. doi:10.1214/15-AIHP674. https://projecteuclid.org/euclid.aihp/1469723516


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