# Zooming-in on a L\'evy process: Failure to observe threshold exceedance over a dense grid

## TL;DR

This paper investigates the asymptotic probability that a Lévy process exceeds a threshold between observations on a dense grid, revealing new insights into the process's behavior under zooming-in limits and regularity conditions.

## Contribution

It establishes the asymptotic behavior of threshold exceedance probabilities for Lévy processes with a zooming-in limit, extending understanding of their fine-scale properties.

## Key findings

- Exact asymptotics for exceedance probability as grid density increases
- Analysis of moments of supremum and maximum differences
- Regularity properties of ladder processes derived

## Abstract

For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/\alpha$-self-similar L\'evy process with $\alpha\in(0,2]$, and restrict to $\alpha>1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.