A wavelet-based almost-sure uniform 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∈[0,1]} of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (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.