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

TL;DR

This paper presents an $L^p$ uniform approximation method for fractional Brownian motion with Hurst exponent $0<H<1/2$, using continuous-time random walks embedded in Brownian motion, with explicit convergence rates.

Contribution

It introduces a novel $L^p$ uniform approximation scheme for fractional Brownian motion with explicit convergence rates based on a pathwise representation.

Findings

Convergence rate is $O( ext{jump size}^{p(1-2 ext{lambda})+ 2( ext{delta}-1)})$.

Applicable for any jump size sequence $ ext{epsilon}_k$ within specified parameters.

Provides a constructive approximation method for fractional Brownian motion with $H<1/2$.

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