On the law of the supremum of Lévy processes

Abstract

We show that the law of the overall supremum ${\bar{X}}_t={\sup }_{s\le t}{X}_s$ of a Lévy process $X$ before the deterministic time $t$ is equivalent to the average occupation measure $μ_t(dx)=\int_0^t\p(X_s\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty ,0)$ and $(0,\infty )$. In this case, $\p(\bar{X}_t\in dx)$ 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 $({\bar{X}}_t,X_t)$, $(g_t,{\bar{X}}_t)$ and $(g_t,{\bar{X}}_t,X_t)$, 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 $\p(g_t\in ds,X_t\in dx,\bar{X}_t\in dy)$ 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.

References in corpus (1)

• Asymptotic behaviour of first passage time distributions for Lévy processes

Cited by in corpus (3)

• The convex minorant of a Lévy process
• Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér-Lundberg and convolution equivalent conditions
• Asymptotic behaviour of first passage time distributions for Lévy processes