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

Author

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  • El Mehdi Haress

    (University of Paris-Saclay)

  • Yaozhong Hu

    (University of Alberta at Edmonton)

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.

Suggested Citation

  • El Mehdi Haress & Yaozhong Hu, 2021. "Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 327-351, July.
  • Handle: RePEc:spr:sistpr:v:24:y:2021:i:2:d:10.1007_s11203-020-09235-z
    DOI: 10.1007/s11203-020-09235-z
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    References listed on IDEAS

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    1. Yaozhong Hu & David Nualart & Hongjuan Zhou, 2019. "Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter," Statistical Inference for Stochastic Processes, Springer, vol. 22(1), pages 111-142, April.
    2. Alexandre Brouste & Stefano Iacus, 2013. "Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package," Computational Statistics, Springer, vol. 28(4), pages 1529-1547, August.
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    Cited by:

    1. John-Fritz Thony & Jean Vaillant, 2022. "Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    2. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.

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