Approximation of fractional Brownian motion by martingales

TL;DR

This paper investigates the optimal way to approximate fractional Brownian motion using martingales, establishing the existence and uniqueness of the best approximation and providing numerical insights.

Contribution

It introduces a unique martingale approximation for fractional Brownian motion and characterizes its specific form, advancing understanding of stochastic process approximations.

Findings

Existence of a unique closest martingale to fractional Brownian motion

Explicit form of the optimal approximating martingale

Numerical results illustrating the approximation quality

Abstract

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.