HAL : hal-00354081, version 2

arXiv : 0901.2808

DOI : 10.1007/978-0-8176-4888-6

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Recent Developments in Fractals and Related Fields (2010) 311--326

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v1 (19-01-2009)

v2 (16-04-2009)

A process very similar to multifractional Brownian motion

Antoine Ayache 1, Pierre Bertrand 2, 3

(2010)

In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is obtained by replacing the constant parameter $H$ of the fractional Brownian motion (fBm) by a smooth enough functional parameter $H(.)$ depending on the time $t$. Here, we consider the process $Z$ obtained by replacing in the wavelet expansion of the fBm the index $H$ by a function $H(.)$ depending on the dyadic point $k/2^j$. This process was introduced in Benassi et al (2000) to model fBm with piece-wise constant Hurst index and continuous paths. In this work, we investigate the case where the functional parameter satisfies an uniform Hölder condition of order $\beta>\sup_{t\in \rit} H(t)$ and ones shows that, in this case, the process $Z$ is very similar to the mBm in the following senses: i) the difference between $Z$ and a mBm satisfies an uniform Hölder condition of order $d>\sup_{t\in \R} H(t)$; ii) as a by product, one deduces that at each point $t\in \R$ the pointwise Hölder exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.

1 :	Laboratoire Paul Painlevé (LPP)
	CNRS : UMR8524 – Université Lille I - Sciences et technologies
2 :	APIS (INRIA Saclay - Ile de France)
	INRIA – Ecole Centrale Paris
3 :	Laboratoire de Mathématiques
	CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II

Domaine

:

Statistiques/Méthodologie Mathématiques/Probabilités

Mots Clés : fractional Brownian motion – wavelet series expansion – multifractional Brownian motion – pointwise Holder regularity.

hal-00354081, version 2

http://hal.archives-ouvertes.fr/hal-00354081

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Contributeur : Pierre, Raphael Bertrand <>

Soumis le : Jeudi 16 Avril 2009, 14:03:52

Dernière modification le : Jeudi 13 Octobre 2011, 10:08:25