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

TL;DR

This paper presents a wavelet-based method for almost sure uniform approximation of fractional Brownian motion using Haar wavelets, along with a parallel algorithm for efficient sample path generation.

Contribution

It introduces a novel wavelet-based approximation of fBm with a proven convergence rate and a parallel algorithm for efficient simulation.

Findings

Approximation achieves almost sure uniform convergence.

Haar wavelets with one vanishing moment suffice for the approximation.

Parallel algorithm enables efficient sample path generation.

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.