Fine regularity of Lévy processes and linear (multi)fractional stable motion

Paul Balança

doi:10.1214/ejp.v19-3393

Abstract

In this work, we investigate the fine regularity of Levy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Levy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

Highlights

The study of sample path continuity and Hölder regularity of stochastic processes is a very active field of research in probability theory
The existing literature provides a variety of uniform results on local regularity, especially on the modulus of continuity, for rather general classes of random fields
We present in particular how this formalism allows to capture and describe the oscillating ejp.ejpecp.org singularities of Lévy processes

Summary

Introduction

The study of sample path continuity and Hölder regularity of stochastic processes is a very active field of research in probability theory. This type of sample path behaviour was first put into light on Brownian motion by Orey and Taylor [39] and Perkins [40] They respectively studied fast and slow points which characterize logarithmic variations of the pointwise modulus of continuity, and proved that the sets of times with a given pointwise regularity have a distinct fractal geometry. The 2-microlocal frontier σf,t(·) offers a richer and more complete description of the local regularity and cover in particular the usual Hölder exponents: αf,t = σf,t(0) and αf,t = − inf{s : σf,t(s ) ≥ 0}, where the last equality has been proved by Meyer [38] under the assumption ω(h) =. We extend this analysis to the multifractional generalisation of the LFSM

Statement of the main results

Lévy processes

Pointwise exponent