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2020 Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid
Krzysztof Bisewski, Jevgenijs Ivanovs
Electron. J. Probab. 25: 1-33 (2020). DOI: 10.1214/20-EJP513


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.


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Krzysztof Bisewski. Jevgenijs Ivanovs. "Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid." Electron. J. Probab. 25 1 - 33, 2020.


Received: 7 June 2019; Accepted: 26 August 2020; Published: 2020
First available in Project Euclid: 21 September 2020

zbMATH: 07252707
MathSciNet: MR4161123
Digital Object Identifier: 10.1214/20-EJP513

Primary: 60G51
Secondary: 60F99


Vol.25 • 2020
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