Variance Estimation for Fractional Brownian Motions with Fixed Hurst Parameters

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter σ², an energy level. This article discusses estimation of a scaling parameter σ² when a Hurst parameter is known. To estimate σ², we propose three approaches based on maximum likelihood estimation, moment-matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. We also justify the improvement of the estimation of σ² when a Hurst parameter is known. Using the three approaches and a parametric bootstrap methodology in a simulation study, we compare the confidence intervals of σ² in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the tested approaches. We found that the approach based on maximum likelihood estimation was optimal in terms of efficiency and accuracy, but computationally expensive. The moment-matching approach was found to be not only comparably efficient and accurate but also computationally fast and robust to deviations from the fractional Brownian motion model.