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Multifractional Lévy Motions

Abstract : In a first part, the class of Real Harmonizable Multifractional Lévy Motions, in short RHMLMs, is introduced. This class is a generalization of the Multifractional Brownian Motion, in short MBM, and of the class of Real Harmonizable Fractional Lévy Motions. This class contains some non-Gaussian second order fields which share many properties with the MBM. Especially, RHMLMs are locally asymptotically self-similar and their pointwise Hölder exponent is allowed to vary along the trajectory. Moreover, their properties are governed by their multifractional function which can be estimated with the localized generalized quadratic variations as in the case of the MBM. The second part deals with the simulation of the non-Gaussian part of a RHMLM. Actually, the method for generating the sample paths of RHMLMs is based on a generalized shot-noise series expansion. However, in some cases, one part of the RHMLM is approximated by a MBM. The last part introduces a locally asymptotically self-similar field $X_(H,\be)$ with a special behaviour at $x=0$. More precisely, at $x\ne 0$, the tangent field is a Fractional Brownian Motion, in short FBM. However, in most cases, the tangent field at $x=0$ is not a FBM and can even be non-Gaussian. In addition, the field $X_(H,\be)$ is asymptotically self-similar at infinity with a Gaussian field, which is not a FBM, as tangent field. Finally, the trajectories regularity and the Hausdorff dimension of the graphs of $X_(H,\be)$ are studied.
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Contributor : Céline Lacaux Connect in order to contact the contributor
Submitted on : Friday, September 17, 2004 - 4:12:47 PM
Last modification on : Tuesday, October 19, 2021 - 11:16:22 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:34:39 PM


  • HAL Id : tel-00006908, version 1


Céline Lacaux. Multifractional Lévy Motions. Mathematics [math]. Université Paul Sabatier - Toulouse III, 2004. English. ⟨tel-00006908⟩



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