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Nonparametric estimation for I.I.D. paths of fractional SDE

Author

Listed:
  • Fabienne Comte

    (Université de Paris)

  • Nicolas Marie

    (Université Paris Nanterre
    ESME Sudria)

Abstract

This paper deals with nonparametric estimators of the drift function b computed from independent continuous observations, on a compact time interval, of the solution of a stochastic differential equation driven by the fractional Brownian motion (fSDE). First, a risk bound is established on a Skorokhod’s integral based least squares oracle $${\widehat{b}}$$ b ^ of b. Thanks to the relationship between the solution of the fSDE and its derivative with respect to the initial condition, a risk bound is deduced on a calculable approximation of $${\widehat{b}}$$ b ^ . Another bound is directly established on an estimator of $$b'$$ b ′ for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the $${\mathbb {R}}$$ R -supported Hermite basis.

Suggested Citation

  • Fabienne Comte & Nicolas Marie, 2021. "Nonparametric estimation for I.I.D. paths of fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 669-705, October.
    • Handle: RePEc:spr:sistpr:v:24:y:2021:i:3:d:10.1007_s11203-021-09246-4
    DOI: 10.1007/s11203-021-09246-4
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    References listed on IDEAS

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    6. Fabienne Comte & Nicolas Marie, 2019. "Nonparametric estimation in fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 359-382, October.
    7. Christophe Denis & Charlotte Dion & Miguel Martinez, 2020. "Consistent procedures for multiclass classification of discrete diffusion paths," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(2), pages 516-554, June.
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    9. Denis Belomestny & Fabienne Comte & Valentine Genon-Catalot, 2019. "Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$ R," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 29-62, February.
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    Cited by:

    1. Nicolas Marie, 2023. "Nonparametric estimation for i.i.d. paths of a martingale-driven model with application to non-autonomous financial models," Finance and Stochastics, Springer, vol. 27(1), pages 97-126, January.
    2. Comte, Fabienne & Marie, Nicolas, 2023. "Nonparametric drift estimation from diffusions with correlated Brownian motions," Journal of Multivariate Analysis, Elsevier, vol. 198(C).

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