# Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires

Abstract : This thesis deals with statistical problems related to two parametric models : the fractional Brownian motion (fBm) and the multifractional Brownian motion (mBm). Firstly, we develop methods for identifying a fBm. These methods are based on the $k$-th absolute empirical moment of discrete variations of a discretized path of a fBm. We then extend this work by developping a robust method (with respect to an additive Gaussian noise) estimation of parameters, and present few tests for validating the model. We also prove a linear algebra result concerning inverses of matrices with terms decreasing hyperbolically far away from the diagonal. This result is fundamental to exhibit the asymptotic behavior of Cramèr-Rao bounds for the parameters of a fBm. Finally, we propose a local approach of the results of the first chapters, in order to identify the mBm, viewed as an extension of the fBm in the sense that the regularity of the process varies with time. Numerical studies and many simulations are provided to illustrate the results of this thesis. Moreover, three appendices allow the reader to familiarize himself with the problem of fBm's identification and the simulation of discretized paths of Gaussian processes.
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https://tel.archives-ouvertes.fr/tel-00006736
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Submitted on : Tuesday, August 24, 2004 - 3:45:04 PM
Last modification on : Friday, November 6, 2020 - 4:09:43 AM
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• HAL Id : tel-00006736, version 1

### Citation

Jean-François Coeurjolly. Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. Modélisation et simulation. Université Joseph-Fourier - Grenoble I, 2000. Français. ⟨tel-00006736⟩

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