Bounds on the Expected Value of Maximum Loss of Fractional Brownian Motion

TL;DR

This paper establishes theoretical bounds on the expected maximum loss of fractional Brownian motion with Hurst parameter between 1/2 and 1, supported by numerical and simulation studies.

Contribution

It provides the first theoretical bounds for the expected maximum loss of fBm in the specified Hurst range, extending to any fixed time and including numerical validation.

Findings

Expected maximum loss bounds: between 1/√π and 2/√π.

Numerical lower bounds obtained via discretization.

Simulation results confirm theoretical bounds.

Abstract

In this study, it is theoretically proven that the expected value of maximum loss of fractional Brownian motion (fBm) up to time 1 with Hurst parameter $[1/2,1)$ is bounded above by $2/\sqrt{\pi}$ and below by $1/\sqrt{\pi}$. This result is generalized for fBm with $H\in[1/2,1)$ up to any fixed time, $t$. This also leads us to the bounds related to the distribution of maximum loss of fBm. As numerical study some lower bounds on the expected value of maximum loss of fBm up to time 1 are obtained by discretization. Simulation study is conducted with Cholesky method. Finally, comparison of the established bounds with simulation results is given.