Zooming in on a Lévy process at its supremum
Zooming in on a Lévy process at its supremum
Let $M$ and $\tau$ be the supremum and its time of a Lévy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $((X_{\tau+t\varepsilon}-M)/a_{\varepsilon})_{t\in\mathbb{R}}$ as $\varepsilon\downarrow0$, results in $(\xi_{t})_{t\in\mathbb{R}}$ constructed from two independent processes having the laws of some …