Definition, properties and wavelet analysis of multiscale fractional Brownian motion

International audienceIn 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 depending 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