A process very similar to multifractional Brownian motion

Abstract

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 $β>\sup_{t\in \rit} H(t)$ 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 $d>{\sup }_{t\in \mathbb{R}}H(t)$; ii) as a by product, one deduces that at each point $t\in \mathbb{R}$ the pointwise Hölder exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.

18 pages

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