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/2j. 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 β>t∈ 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>t∈ H(t); ii) as a by product, one deduces that at each point t∈ the pointwise H\"older exponent of Z is H(t) and that Z is tangent to a fBm with Hurst parameter H(t).