$L^p$ uniform random walk-type approximation for fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$

Abstract

In this note, we prove an $L^p$ uniform approximation of the fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice $\epsilon_k$ for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is $O(\epsilon_k^{p(1-2\lambda)+ 2(\delta-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}< \delta < 1$, $\lambda \in \big(\frac{1-H}{2}, \frac{1}{2} + \frac{\delta-1}{p}\big)$.