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Estimation of several parameters in discretely-observed stochastic differential equations with additive fractional noise

Author

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  • El Mehdi Haress

    (Université Paris-Saclay)

  • Alexandre Richard

    (Université Paris-Saclay)

Abstract

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein–Uhlenbeck processes and illustrate our results with some numerical experiments.

Suggested Citation

  • El Mehdi Haress & Alexandre Richard, 2024. "Estimation of several parameters in discretely-observed stochastic differential equations with additive fractional noise," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 641-691, October.
    • Handle: RePEc:spr:sistpr:v:27:y:2024:i:3:d:10.1007_s11203-024-09311-8
    DOI: 10.1007/s11203-024-09311-8
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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. Cohen, Serge & Panloup, Fabien, 2011. "Approximation of stationary solutions of Gaussian driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2776-2801.
    3. Kubilius, K. & Mishura, Y., 2012. "The rate of convergence of Hurst index estimate for the stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3718-3739.
    4. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    5. Maria Jolis & Noèlia Viles, 2007. "Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 20(2), pages 133-152, June.
    6. 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.
    7. Jolis, Maria & Viles, Noèlia, 2007. "Continuity with respect to the Hurst parameter of the laws of the multiple fractional integrals," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1189-1207, September.
    8. Giordano, Luca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2020. "SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7396-7430.
    9. Richard, Alexandre, 2015. "A fractional Brownian field indexed by L2 and a varying Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1394-1425.
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