Definition, properties and wavelet analysis of multiscale fractional Brownian motion

Jean-Marc Bardet (1, 2) , Pierre, Raphael Bertrand (3)

1 SAMOS - Statistique Appliquée et MOdélisation Stochastique
2 CES - Centre d'économie de la Sorbonne
3 PAS

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 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.

Mots clés: Fractional Brownian motion; Long-range dependence; Path regularity; Self-similarity; Wavelet analysis; Functional Central Limit Theorem

Domaines: Statistiques [math.ST]

Jean-Marc Bardet, Pierre, Raphael Bertrand. Definition, properties and wavelet analysis of multiscale fractional Brownian motion. Fractals, 2007, 15 (1), pp.73-87. ⟨hal-00127938⟩