Estimation Of all parameters in the Fractional Ornstein-Uhlenbeck model under discrete observations

04/10/2020

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by   El Mehdi Haress, et al.

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Let the Ornstein-Uhlenbeck process (X_t)_t>0 driven by a fractional Brownian motion B^H, described by dX_t = -θ X_t dt + σ dB_t^H be observed at discrete time instants t_k=kh, k=0, 1, 2, ..., 2n+2. We propose ergodic type statistical estimators θ̂_n, Ĥ_n and σ̂_n to estimate all the parameters θ, H and σ in the above Ornstein-Uhlenbeck model simultaneously. We prove the strong consistence and the rate of convergence of the estimators. 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.