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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 Levy 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… Click to show full abstract

For a Levy 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 Levy 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.

Keywords: threshold exceedance; process failure; exceedance dense; failure observe; observe threshold; zooming process

Journal Title: Electronic Journal of Probability
Year Published: 2020

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