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

Abstract

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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Cited by in corpus (7)

• Geometrically Convergent Simulation of the Extrema of Lévy Processes
• Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
• Optimal estimation of some random quantities of a Lévy process
• On the speed of convergence of discrete Pickands constants to continuous ones
• Monte Carlo algorithm for the extrema of tempered stable processes
• Intermittency in the small-time behavior of Lévy processes
• Local behavior of diffusions at the supremum