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Parameter estimation for the Langevin equation with stationary-increment Gaussian noise

Author

Listed:
  • Tommi Sottinen

    (University of Vaasa)

  • Lauri Viitasaari

    (Aalto University School of Science, Helsinki)

Abstract

We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry–Esseen bound of the so-called second moment estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein–Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.

Suggested Citation

  • Tommi Sottinen & Lauri Viitasaari, 2018. "Parameter estimation for the Langevin equation with stationary-increment Gaussian noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 569-601, October.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:3:d:10.1007_s11203-017-9156-6
    DOI: 10.1007/s11203-017-9156-6
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    References listed on IDEAS

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    1. Sottinen, Tommi & Yazigi, Adil, 2014. "Generalized Gaussian bridges," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3084-3105.
    2. Sun, Xiaoxia & Guo, Feng, 2015. "On integration by parts formula and characterization of fractional Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 170-177.
    3. Yazigi, Adil, 2015. "Representation of self-similar Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 94-100.
    4. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    5. Tommi Sottinen & Lauri Viitasaari, 2016. "Stochastic Analysis of Gaussian Processes via Fredholm Representation," International Journal of Stochastic Analysis, Hindawi, vol. 2016, pages 1-15, July.
    6. Tommi Sottinen & Ciprian Tudor, 2008. "Parameter estimation for stochastic equations with additive fractional Brownian sheet," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 221-236, October.
    7. Katsuto Tanaka, 2015. "Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 315-332, October.
    8. Azmoodeh, Ehsan & Sottinen, Tommi & Viitasaari, Lauri & Yazigi, Adil, 2014. "Necessary and sufficient conditions for Hölder continuity of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 230-235.
    9. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 205-227, October.
    10. Viitasaari, Lauri, 2016. "Representation of stationary and stationary increment processes via Langevin equation and self-similar processes," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 45-53.
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    Cited by:

    1. Pauliina Ilmonen & Soledad Torres & Lauri Viitasaari, 2020. "Oscillating Gaussian processes," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 571-593, October.
    2. Rachid Belfadli & Khalifa Es-Sebaiy & Fatima-Ezzahra Farah, 2022. "Statistical analysis of the non-ergodic fractional Ornstein–Uhlenbeck process with periodic mean," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 885-911, October.
    3. Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.

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