A wavelet-based approximation of fractional Brownian motion with a parallel algorithm

Abstract: We construct a wavelet-based almost sure uniform approximation of fractional Brownian motion (fBm) B_t^H, t in [0, 1], of Hurst index H in (0, 1). Our results show that by Haar wavelets which merely have one vanishing moment, an almost sure uniform expansion of fBm of H in (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an fBm efficiently.