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

• Main Author: Vitalii Makogin
• Format: Article
• Language: English
• Published: VTeX 2016-07-01
• Series: Modern Stochastics: Theory and Applications
• Subjects: fractional Brownian motion; Monte Carlo simulations; expected maximum; discrete approximation
• Online Access: https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA59

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 ∞ as the Hurst index tends to 0.