# On the law of the supremum of Lévy processes

Abstract : We show that the law of the overall supremum $\overline{X}_t=\sup_{s\le t}X_s$ of a Lévy process $X$ before the deterministic time $t$ is equivalent to the average occupation measure $\mu_t(dx)=\int_0^t\p(X_s\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, $\p(\overline{X}_t\in dx)$ is absolutely continuous for some (and hence for all) $t>0$, if and only if the resolvent measure of $X$ is absolutely continuous. We also study the cases where 0 is not regular for one of the halflines $(-\infty,0)$ or $(0,\infty)$. Then we give absolute continuity criterions for the laws of $(\overline{X}_t,X_t)$, $(g_t,\overline{X}_t)$ and $(g_t,\overline{X}_t,X_t)$, where $g_t$ is the time at which the supremum occurs before $t$. The proofs of these results use an expression of the joint law $\p(g_t\in ds,X_t\in dx,\overline{X}_t\in dy)$ in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made (partly) explicit in some particular instances.
Mots-clés :
Type de document :
Pré-publication, Document de travail
2010
Domaine :

https://hal.archives-ouvertes.fr/hal-00537222
Contributeur : Loïc Chaumont <>
Soumis le : jeudi 30 mai 2013 - 21:51:00
Dernière modification le : mercredi 19 décembre 2018 - 14:08:04
Document(s) archivé(s) le : samedi 31 août 2013 - 08:55:07

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aop708.pdf
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### Identifiants

• HAL Id : hal-00537222, version 2
• ARXIV : 1011.4151

### Citation

Loïc Chaumont. On the law of the supremum of Lévy processes. 2010. 〈hal-00537222v2〉

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