Simulation paradoxes related to a fractional Brownian motion with small Hurst index

TL;DR

This paper investigates the challenges in simulating fractional Brownian motion with small Hurst index, revealing that approximation errors grow unbounded as the Hurst index approaches zero, highlighting paradoxical simulation behavior.

Contribution

It provides a theoretical analysis of the error growth in simulating fractional Brownian motion with small Hurst index, demonstrating a divergence in expected maximum approximation error.

Findings

Approximation error increases rapidly as Hurst index approaches zero.

Error of the expected maximum diverges to infinity for fixed sample size.

Simulation paradoxes are identified for small Hurst index values.

Abstract

We consider the simulation of sample paths of a fractional Brownian motion with small values of the Hurst index and estimate the behavior of the expected maximum. We prove that, for each fixed $N$, the error of approximation $\mathbf {E}\max_{t\in[0,1]}B^H(t)-\mathbf {E}\max_{i=\overline{1,N}}B^H(i/N)$ grows rapidly to $\infty$ as the Hurst index tends to 0.