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Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. (English) Zbl 0984.62058
Summary: This paper develops a class of consistent estimators of the parameters of a fractional Brownian motion based on the asymptotic behavior of the $k$-th absolute moment of discrete variations of its sampled paths over a discrete grid of the interval $\left[0,1\right]$. We derive explicit convergence rates for these types of estimators, valid through the whole range $0 of the self-similarity parameter. We also establish the asymptotic normality of our estimators. The effectiveness of our procedure is investigated in a simulation study.
##### MSC:
 62M05 Markov processes: estimation 62G20 Nonparametric asymptotic efficiency 62G05 Nonparametric estimation 62M10 Time series, auto-correlation, regression, etc. (statistics) 60J65 Brownian motion