Zooming in on a L\'evy process at its supremum

TL;DR

This paper investigates the local behavior of Lévy processes at their supremum, revealing that zooming in yields a process constructed from self-similar Lévy processes conditioned to stay positive or negative, with applications to discretization error analysis.

Contribution

It introduces a novel zooming-in approach at the supremum of Lévy processes, contrasting with classical zooming out, and characterizes the domains of attraction for this procedure.

Findings

Zooming in at the supremum leads to a process from two independent self-similar Lévy processes.

Establishes a limit theorem for discretization errors in supremum simulation.

Provides a complete characterization of attraction domains for zooming in at zero.

Abstract

Let $M$ and $\tau$ be the supremum and its time of a L\'evy 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 $\varepsilon\downarrow 0$, results in $(\xi_t)_{t\in\mathbb R}$ constructed from two independent processes having the laws of some self-similar L\'evy 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 of Lamperti (1962). 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 of Asmussen, Glynn and Pitman (1995) for the Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a L\'evy process at 0 is provided.