Upper functions for sample paths of L\'evy(-type) processes

TL;DR

This paper investigates the small-time behavior of sample paths of Lévy and Lévy-type processes, establishing criteria for their limsup behavior and introducing a new maximal inequality.

Contribution

It provides integral criteria for the small-time asymptotics of Lévy-type processes and introduces a novel maximal inequality for these processes.

Findings

Criteria for finiteness of limsup of scaled process paths

Extension of classical results for Lévy processes

A new maximal inequality for Lévy-type processes

Abstract

We study the small-time asymptotics of sample paths of L\'evy processes and L\'evy-type processes. Namely, we investigate under which conditions the limit $$\limsup_{t \to 0} \frac{1}{f(t)} |X_t-X_0|$$ 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 L\'evy-driven SDEs and stable-like processes. For the particular case of L\'evy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for L\'evy-type processes, which is of independent interest.