Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations

Let the Ornstein–Uhlenbeck process $$(X_t)_{t\ge 0}$$ ( X t ) t ≥ 0 driven by a fractional Brownian motion $$B^{H }$$ B H described by $$dX_t = -\theta X_t dt + \sigma dB_t^{H }$$ d X t = - θ X t d t + σ d B t H be observed at discrete time instants $$t_k=kh$$ t k = k h , $$k=0, 1, 2, \ldots , 2n+2 $$ k = 0 , 1 , 2 , … , 2 n + 2 . We propose an ergodic type statistical estimator $${\hat{\theta }}_n $$ θ ^ n , $${\hat{H}}_n $$ H ^ n and $${\hat{\sigma }}_n $$ σ ^ n to estimate all the parameters $$\theta $$ θ , H and $$\sigma $$ σ in the above Ornstein–Uhlenbeck model simultaneously. We prove the strong consistence and the rate of convergence of the estimator. The step size h can be arbitrarily fixed and will not be forced to go zero, which is usually a reality. The tools to use are the generalized moment approach (via ergodic theorem) and the Malliavin calculus.