Fine regularity of Lévy processes and linear (multi)fractional stable motion
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Publication:2514293
DOI10.1214/EJP.V19-3393zbMATH Open1307.60055arXiv1302.3140MaRDI QIDQ2514293
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Publication date: 3 February 2015
Published in: (Search for Journal in Brave)
Abstract: In this work, we investigate the fine regularity of L'evy 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 L'evy 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 {alpha}-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.
Full work available at URL: https://arxiv.org/abs/1302.3140
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