$L^{p}$ 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 $\max \{0,1-\frac {pH}{2}\}< \delta < 1$, $\lambda \in \big (\frac {1-H}{2}, \frac {1}{2} + \frac {\delta -1}{p}\big )$.