The Annals of Probability

On the law of the supremum of Lévy processes

L. Chaumont

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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

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

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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