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2014 Fine regularity of Lévy processes and linear (multi)fractional stable motion
Paul Balança
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Electron. J. Probab. 19: 1-37 (2014). DOI: 10.1214/EJP.v19-3393

Abstract

In this work, we investigate the fine regularity of Lévy 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 Lévy 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.

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Paul Balança. "Fine regularity of Lévy processes and linear (multi)fractional stable motion." Electron. J. Probab. 19 1 - 37, 2014. https://doi.org/10.1214/EJP.v19-3393

Information

Accepted: 26 October 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60055
MathSciNet: MR3275853
Digital Object Identifier: 10.1214/EJP.v19-3393

Subjects:
Primary: 60G07
Secondary: 60G17 , 60G22 , 60G44

Keywords: 2-microlocal analysis , Hölder regularity , Lévy processes , linear fractional stable motion , multifractal spectrum , oscillating singularities

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