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

2021/02/12 by Franziska Kühn, Kühn, Franziska
Economics, Econometrics and Finance · #47G20 #60G17 #60G51 #60G53 #60J76 #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2102.06541

openalex publication_date 2021/02/12 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28

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 \limsup_{t → 0} (1)/(f(t)) |X_t-X₀| 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.

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