On tails of exit times of multidimensional Lévy processes

Abstract

Using a very simple argument based on the indepenence of increments and the fact that in a finite dimensional space $R^d$ there are not too many directions, we derive a theorem stating that exit time of any (non-constant) Lévy process on $R^d$ from a ball has exponentially light tails.

Estimates with better constants may be obtained from the estimate in Proposition 5.2 in: "Asymptotic Behaviour and Estimates of Slowly Varying Convolution Semigroups" by Tomasz Grzywny, Michał Ryznar and Bartosz Trojan