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

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 $H\in (0,\frac{1}{2})$. 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 $H=\frac{1}{2}$ in the sense of recent work by Bisewski, Dębicki & Rolski (2021).

23 pages, 3 figures