Upper functions for sample paths of Lévy(-type) processes

Abstract

We study the small-time asymptotics of sample paths of Lévy processes and Lévy-type processes. Namely, we investigate under which conditions the limit

$$\underset{t\to 0}{lim\hspace{0.17em}sup}\frac{1}{f(t)}|X_t-X_0|$$

is finite resp. infinite with probability 1. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to Lévy-driven SDEs and stable-like processes. For the particular case of Lévy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for Lévy-type processes, which is of independent interest.