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Consistency of the drift parameter estimator for the discretized fractional Ornstein-Uhlenbeck process with Hurst index \(H\in(0,\frac{1}{2})\). (English) Zbl 1326.60048
Summary: We consider the Langevin equation which contains an unknown drift parameter \(\theta\) and where the noise is modeled as a fractional Brownian motion with Hurst index \(H\in(0,\frac{1}{2})\). The solution corresponds to the fractional Ornstein-Uhlenbeck process. We construct an estimator, based on discrete observations in time, of the unknown drift parameter that is similar in form to the maximum likelihood estimator for the drift parameter in the Langevin equation with standard Brownian motion. It is assumed that the interval between observations is \(n^{-1}\), i.e. tends to zero (high-frequency data) and the number of observations increases to infinity as \(n^{m}\) with \(m>1\). It is proved that for strictly positive \(\theta\) the estimator is strongly consistent for any \(m>1\), while for \(\theta\leq0\) it is consistent when \(m>\frac{1}{2H}\).

60G22 Fractional processes, including fractional Brownian motion
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
60F15 Strong limit theorems
60F25 \(L^p\)-limit theorems
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