The Multifractal Nature of Volterra-L\'{e}vy Processes

TL;DR

This paper investigates the multifractal properties and regularity of sample paths of Volterra-Lévy processes, revealing their singularity spectrum and 2-microlocal frontier under certain conditions.

Contribution

It provides the first detailed analysis of the multifractal spectrum and local regularity of Volterra-Lévy processes based on the properties of the kernel function.

Findings

Derived the spectrum of singularities for Volterra-Lévy processes

Established the 2-microlocal frontier under regularity assumptions

Connected the regularity of the process to the properties of the kernel function

Abstract

We consider the regularity of sample paths of Volterra-L\'{e}vy processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a L\'{e}vy process and $F$ is a deterministic real-valued function. We derive the spectrum of singularities and a result on the 2-microlocal frontier of $\{M(t)\}_{t\in [0,1]}$, under regularity assumptions on the function $F$.