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May 2013 On the law of the supremum of Lévy processes
L. Chaumont
Ann. Probab. 41(3A): 1191-1217 (May 2013). DOI: 10.1214/11-AOP708

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.

Citation

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L. Chaumont. "On the law of the supremum of Lévy processes." Ann. Probab. 41 (3A) 1191 - 1217, May 2013. https://doi.org/10.1214/11-AOP708

Information

Published: May 2013
First available in Project Euclid: 29 April 2013

zbMATH: 1277.60081
MathSciNet: MR3098676
Digital Object Identifier: 10.1214/11-AOP708

Subjects:
Primary: 60G51

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3A • May 2013
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