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

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Contributors

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 [formula presented] 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.

Details

Original languageEnglish
Pages (from-to)2874-2908
Number of pages35
JournalBernoulli
Volume28
Issue number4
Publication statusPublished - Nov 2022
Peer-reviewedYes

Keywords

ASJC Scopus subject areas

Keywords

  • Feller process, Lévy process, martingale problem, maximal inequality, sample path behaviour, small-time asymptotics, upper function

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