Zooming in on a Lévy process at its supremum

Abstract

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 self-similar Lévy process $\widehat{X}$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $\widehat{X}$ under the zooming-in procedure as opposed to the classical zooming out [Trans. Amer. Math. Soc. 104 (1962) 62–78]. As an application of this result, we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result in [Ann. Appl. Probab. 5 (1995) 875–896] for a linear Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process is provided.