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.

Published at http://dx.doi.org/10.15559/16-VMSTA59 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)