Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index H ∈ (0, 12)

parameter θ and where the noise is modeled as fractional Brownian motionwith Hurst index H ∈ (0, 12 ). The solution corresponds to the fractionalOrnstein–Uhlenbeck process. We construct an estimator, based on discreteobservations in time, of the unknown drift parameter, that is similar in formto the maximum likelihood estimator for the drift parameter in Langevinequation with standard Brownian motion. It is assumed that the intervalbetween observations is n−1, i.e. tends to zero (high-frequency data) andthe number of observations increases to infinity as nm with m > 1. It isproved that for strictly positive θ the estimator is strongly consistent forany m > 1, while for θ ≤ 0 it is consistent when m > 12H