On Fractional Brownian Motion and Wavelets

Given a fractional Brownian motion (fBm) with Hurst index H ? (0,1){H\in(0,1)} , we associate with this a special family of representations of Cuntz algebras related to frequency domains and wavelets. Vice versa, we consider a pair of Haar wavelets satisfying some compatibility conditions, and we construct the covariance functions of fBm with a fixed Hurst index. The Cuntz algebra representations enter the picture as filters of the associated wavelets. Extensions to q-dependent covariance functions leading to a corresponding fBm process will also be discussed.