Year: 2020
Type: article
Abstract: For a Lévy 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 Lévy 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.
DOI: 10.1214/20-ejp513
Authors: Krzysztof Bisewski, Jevgenijs Ivanovs
Institutions: Centrum Wiskunde & Informatica, Aarhus University
Language: English
Topic: Probability and Risk Models
Field: Decision Sciences
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $675
Award: G6070997171
OpenAlex ID: W2936639521