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

El Mehdi Haress () and Yaozhong Hu ()
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El Mehdi Haress: University of Paris-Saclay
Yaozhong Hu: University of Alberta at Edmonton

Statistical Inference for Stochastic Processes, 2021, vol. 24, issue 2, No 3, 327-351

Abstract: Abstract 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.

Keywords: Fractional Brownian motion; Fractional Ornstein–Uhlenbeck; Parameter estimation; Malliavin calculus; Ergodicity; Stationary processes; Newton method; Central limit theorem; 62M09; 60G22; 60H10; 60H30 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s11203-020-09235-z

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