Definition, Properties and Wavelet Analysis of Multiscale Fractional Brownian Motion

In some applications, for instance, finance, biomechanics, turbulence or internet traffic, it is relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is dependent on the frequency. In this contribution, we describe the multiscale fractional Brownian motions which present a parameter H as a piecewise constant function of the frequency. We provide the main properties of these processes: long-memory and smoothness of the paths. Then we propose a statistical method based on wavelet analysis to estimate the different parameters and prove a functional Central Limit Theorem satisfied by the empirical variance of the wavelet coefficients.