# The asymptotic behavior of the density of the supremum of Lévy processes

* Auteur correspondant
Abstract : Let us consider a real 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.
Keywords :
Type de document :
Pré-publication, Document de travail
2013
Domaine :

Littérature citée [18 références]

https://hal.archives-ouvertes.fr/hal-00870233
Contributeur : Loïc Chaumont <>
Soumis le : dimanche 6 octobre 2013 - 16:05:18
Dernière modification le : mercredi 19 décembre 2018 - 14:08:04
Document(s) archivé(s) le : mardi 7 janvier 2014 - 04:26:31

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• HAL Id : hal-00870233, version 1

### Citation

Loïc Chaumont, Jacek Malecki. The asymptotic behavior of the density of the supremum of Lévy processes. 2013. 〈hal-00870233〉

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