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Zooming-in on a lévy process: Failure to observe threshold exceedance over a dense grid

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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/α-self-similar Lévy process with α ∈ (0, 2], and restrict to α > 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.

Original languageEnglish
Article number113
JournalElectronic Journal of Probability
Number of pages33
Publication statusPublished - 2020

    Research areas

  • Discretization error, High frequency, Scaling limits, Small-time behavior, Supremum

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