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We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2 ). Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter θ. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. 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 positive θ the estimator is strongly consistent for any m > 1 and for negative θ it is consistent when m > 1 2H .60G22, 60F15, 60F25, 62F10, 62F12.

Orey suggested the definition of an index for Gaussian process with stationary increments which determines various properties of the sample paths of this process. We provide an extension of the definition of the Orey index towards a second order stochastic process which may not have stationary increments and estimate the Orey index towards a Gaussian process from discrete observations of its sample paths.

Strongly consistent and asymptotically normal estimators of the Hurst parameter of solutions of stochastic differential equations are proposed. The estimators are based on discrete observations of the underlying processes. IntroductionRecently long range dependence (LRD) became one of the most researched phenomena in statistics. It appears in various applied fields and inspires new models to account for it. Stochastic differential equations (SDEs) are widely used to model continuous time processes. Within this framework, LRD is frequently modeled with the help of SDEs driven by a fractional Brownian motion (fBm). It is well known that the latter Gaussian process is governed by a single parameter H ∈ (0, 1) (called the Hurst index) and that values of H in (1/2, 1) correspond to LRD models. In applications, the estimation of H is a fundamental problem. Its solution depends on the theoretical structure of a model under consideration. Therefore, particular models usually deserve separate analysis. In this paper, we concentrate on the estimation of H under the assumption that an observable continuous time process (Xt) t∈[0,T ] satisfies SDE Xt = ξ + t