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

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