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Large deviations for the Ornstein–Uhlenbeck process without tears

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  • Bercu, Bernard
  • Richou, Adrien

Abstract

Our goal is to establish large deviations for the maximum likelihood estimator of the drift parameter of the Ornstein–Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. It can also be successfully implemented for more complex diffusion processes.

Suggested Citation

  • Bercu, Bernard & Richou, Adrien, 2017. "Large deviations for the Ornstein–Uhlenbeck process without tears," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 45-55.
    • Handle: RePEc:eee:stapro:v:123:y:2017:i:c:p:45-55
    DOI: 10.1016/j.spl.2016.11.030
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    References listed on IDEAS

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    1. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    2. Zani, Marguerite, 2002. "Large deviations for squared radial Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 25-42, November.
    3. Bercu, Bernard & Coutin, Laure & Savy, Nicolas, 2012. "Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3393-3424.
    4. Demni, N. & Zani, M., 2009. "Large deviations for statistics of the Jacobi process," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 518-533, February.
    5. Zhao, Shoujiang & Gao, Fuqing, 2010. "Large deviations in testing Jacobi model," Statistics & Probability Letters, Elsevier, vol. 80(1), pages 34-41, January.
    6. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
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    Cited by:

    1. Zhao, Shoujiang & Liu, Qiaojing, 2020. "A large deviation result for maximum likelihood estimator of non-homogeneous Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 162(C).
    2. Hui Jiang & Qingshan Yang, 2022. "Moderate Deviations for Drift Parameter Estimations in Reflected Ornstein–Uhlenbeck Process," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1262-1283, June.
    3. Gajda, J. & Wyłomańska, A. & Kantz, H. & Chechkin, A.V. & Sikora, G., 2018. "Large deviations of time-averaged statistics for Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 143(C), pages 47-55.
    4. Zhao, Shoujiang & Liu, Qiaojing & Chen, Ting, 2018. "On the large deviation principle for maximum likelihood estimator of α-Brownian bridge," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 143-150.

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