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Parameter Estimation for Fractional Ornstein–Uhlenbeck Processes with Discrete Observations

  • Yaozhong HuEmail author
  • Jian Song
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

Consider an Ornstein–Uhlenbeck process, \(\mathrm{d}X_{t} = -\theta X_{t}\mathrm{d}t + \sigma \mathrm{d}B_{t}^{H}\), driven by fractional Brownian motion B H with known Hurst parameter \(H \geq\frac{1} {2}\) and known variance σ. But the parameter θ > 0 is unknown. Assume that the process is observed at discrete time instants t = h, 2h, , nh. We construct an estimator \(\hat{\theta }_{n}\) of θ which is strongly consistent, namely, \(\hat{\theta }_{n}\) converges to θ almost surely as n → . We also obtain a central limit type theorem and a Berry–Esseen type theorem for this estimator \(\hat{\theta }_{n}\) when \(1/2 \leq H < 3/4\). The tool we use is some recent results on central limit theorems for multiple Wiener integrals through Malliavin calculus. It should be pointed out that no condition on the step size h is required, contrary to the existing conventional assumptions.

Keywords

Central Limit Theorem Fractional Brownian Motion Normal Random Variable Strong Consistency Hurst Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We appreciate Chihoon Lee and the referee’s careful reading of this paper. Yaozhong Hu is partially supported by a grant from the Simons Foundation #209206.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

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