Lower bound for the expected supremum of fractional Brownian motion using coupling

TL;DR

This paper introduces a new theoretical lower bound for the expected maximum of drifted fractional Brownian motion, demonstrating improved accuracy over Monte Carlo methods for certain Hurst indices, and provides new representations and derivatives related to fractional Brownian motion.

Contribution

The paper presents a novel lower bound for the expected supremum of fractional Brownian motion and derives explicit formulas and representations, advancing theoretical understanding and computational methods.

Findings

Lower bound outperforms Monte Carlo estimates for H<1/2

Derived Paley-Wiener-Zygmund representation of linear fractional Brownian motion

Explicit derivative of expected supremum at H=1/2

Abstract

We derive a new theoretical lower bound for the expected supremum of drifted fractional Brownian motion with Hurst index $H\in(0,1)$ over (in)finite time horizon. Extensive simulation experiments indicate that our lower bound outperforms the Monte Carlo estimates based on very dense grids for $H\in(0,\tfrac{1}{2})$. Additionally, we derive the Paley-Wiener-Zygmund representation of a Linear Fractional Brownian motion and give an explicit expression for the derivative of the expected supremum at $H=\tfrac{1}{2}$ in the sense of recent work by Bisewski, D\k{e}bicki & Rolski (2021).