Distance from fractional Brownian motion with associated Hurst index $0<H<1/2$ to the subspaces of Gaussian martingales involving power integrands with an arbitrary positive exponent

TL;DR

This paper investigates the optimal approximation of fractional Brownian motion with Hurst index less than 1/2 by specific Gaussian martingales involving power integrands, providing insights into their geometric relationships.

Contribution

It introduces a novel approach to approximate fractional Brownian motion with Hurst index in (0,1/2) using Gaussian martingales with power integrands, expanding understanding of their proximity.

Findings

Derived the best approximation criteria for fractional Brownian motion

Characterized the subspace of Gaussian martingales involving power integrands

Enhanced understanding of the geometric structure of fractional Brownian motion

Abstract

We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form $\int _0^ts^{\gamma}dW_s$, where $W$ is a Wiener process, $\gamma >0$.