A process very similar to multifractional Brownian motion

TL;DR

This paper compares a wavelet-based process with multifractional Brownian motion, showing they are very similar under certain regularity conditions, and analyzes their local regularity properties.

Contribution

It demonstrates that a wavelet-based process with a variable index closely approximates multifractional Brownian motion when the index function is sufficiently regular.

Findings

The process Z is similar to mBm under a Hölder condition.

The pointwise Hölder exponent of Z equals H(t).

Z is tangent to a fBm with Hurst parameter H(t).

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 H\"older 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 H\"older 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 H\"older exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.