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

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.