The Annals of Probability

On the law of the supremum of Lévy processes

L. Chaumont

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Abstract

We show that the law of the overall supremum $\overline{X}_{t}=\sup_{s\let}X_{s}$ of a Lévy process $X$, before the deterministic time $t$ is equivalent to the average occupation measure $\mu_{t}^{+}(dx)=\int_{0}^{t}\mathbb{P} (X_{s}\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, $\mathbb{P} (\overline{X}_{t}\in dx)$ 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 both halflines. Then we give absolute continuity criterions for the laws of $(g_{t},\overline{X}_{t})$ 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 $\mathbb{P} (g_{t}\in ds,X_{t}\in dx,\overline{X}_{t}\in dy)$ 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.

Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1191-1217.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1367241498

Digital Object Identifier
doi:10.1214/11-AOP708

Mathematical Reviews number (MathSciNet)
MR3098676

Zentralblatt MATH identifier
1277.60081

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

Keywords
Past supremum equivalent measures absolute continuity average occupation measure reflected process excursion measure

Citation

Chaumont, L. On the law of the supremum of Lévy processes. Ann. Probab. 41 (2013), no. 3A, 1191--1217. doi:10.1214/11-AOP708. https://projecteuclid.org/euclid.aop/1367241498


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