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Oracle inequalities for the stochastic differential equations

Author

Listed:
  • E. A. Pchelintsev

    (Tomsk State University)

  • S. M. Pergamenshchikov

    (University of Rouen
    Tomsk State University
    National Research University “MPEI”)

Abstract

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi-martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein–Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.

Suggested Citation

  • E. A. Pchelintsev & S. M. Pergamenshchikov, 2018. "Oracle inequalities for the stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 469-483, July.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:2:d:10.1007_s11203-018-9180-1
    DOI: 10.1007/s11203-018-9180-1
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    References listed on IDEAS

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    1. Fourdrinier, Dominique & Strawderman, William E., 1996. "A Paradox Concerning Shrinkage Estimators: Should a Known Scale Parameter Be Replaced by an Estimated Value in the Shrinkage Factor?," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 109-140, November.
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    3. Victor Konev & Serguei Pergamenchtchikov, 2010. "General model selection estimation of a periodic regression with a Gaussian noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1083-1111, December.
    4. V. Konev & S. Pergamenshchikov, 2003. "Sequential Estimation of the Parameters in a Trigonometric Regression Model with the Gaussian Coloured Noise," Statistical Inference for Stochastic Processes, Springer, vol. 6(3), pages 215-235, October.
    5. L. Galtchouk & S. Pergamenshchikov, 2009. "Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(1), pages 1-18.
    6. D. Fourdrinier & S. Pergamenshchikov, 2007. "Improved Model Selection Method for a Regression Function with Dependent Noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 435-464, September.
    7. {L}ukasz Delong & Claudia Kluppelberg, 2008. "Optimal investment and consumption in a Black--Scholes market with L\'evy-driven stochastic coefficients," Papers 0806.2570, arXiv.org.
    8. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    9. Galtchouk, L. & Pergamenshchikov, S., 2006. "Asymptotically efficient estimates for nonparametric regression models," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 852-860, April.
    10. Evgeny Pchelintsev, 2013. "Improved estimation in a non-Gaussian parametric regression," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 15-28, April.
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    Cited by:

    1. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Leshchinskaya, 2022. "Improved estimation method for high dimension semimartingale regression models based on discrete data," Statistical Inference for Stochastic Processes, Springer, vol. 25(3), pages 537-576, October.
    2. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Povzun, 2022. "Efficient estimation methods for non-Gaussian regression models in continuous time," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 113-142, February.

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