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For a L\'evy process $X$ on a finite time interval consider the probabilitythat it exceeds some fixed threshold $x>0$ while staying below $x$ at thepoints of a regular grid. We establish exact asymptotic behavior of thisprobability 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\'evyprocess with $\alpha\in(0,2]$, and restrict to $\alpha>1$. Moreover, themoments of the difference of the supremum and the maximum over the grid pointsare analyzed and their asymptotic behavior is derived. It is also shown thatthe zooming-in assumption implies certain regularity properties of the ladderprocess, and the decay rate of the left tail of the supremum distribution isdetermined.

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