Abstract
This paper deals with a Skorokhod’s integral based projection type estimator \({\widehat{b}}_m\) of the drift function \(b_0\) computed from \(N\in \mathbb N^*\) independent copies \(X^1,\dots ,X^N\) of the solution X of \(dX_t = b_0(X_t)dt +\sigma dB_t\), where B is a fractional Brownian motion of Hurst index \(H\in (1/2,1)\). Skorokhod’s integral based estimators cannot be calculated directly from \(X^1,\dots ,X^N\), but in this paper an \(\mathbb L^2\)-error bound is established on a calculable approximation of \({\widehat{b}}_m\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Comte F, Genon-Catalot V (2020) Nonparametric drift estimation for I.I.D. paths of stochastic differential equations. Ann. Stat. 48(6):3336–3365
Comte F, Marie N (2019) Nonparametric estimation in fractional SDE. Stat. Inference Stoch Process 22(3):359–382
Comte F, Marie N (2021) Nonparametric estimation for I.I.D. paths of fractional SDE. Stat Inference Stoch Process 24(3):669–705
Decreusefond L (2022) Selected topics in Malliavin calculus. Bocconi & Springer Series, Berlin
DeVore RA, Lorentz GG (1993) Constructive approximation. Springer, Berlin
Hairer M, Ohashi A (2007) Ergodic theory for SDEs with extrinsic memory. Ann Probab 35(5):1950–1977
Hu Y, Nualart D (2010) Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat Probab Lett 80:1030–1038
Hu Y, Nualart D, Zhou H (2019) Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion. Stochastics 91(8):1067–1091
Kleptsyna ML, Le Breton A (2001) Some explicit statistical results about elementary fractional type models. Nonlinear Anal 47:4783–4794
Kubilius K, Mishura Y, Ralchenko K (2017) Parameter estimation in fractional diffusion models. Springer, Berlin
Li X-M, Panloup F, Sieber J (2023) On the (non-)stationary density of fractional-driven stochastic differential equations. Ann Probab 51(6):2056–2085
Marie N (2023) On a computable Skorokhod’s integral based estimator of the drift parameter in fractional SDE. Preprint arXiv: 2301.05341
Marie N, Raynaud de Fitte P (2021) Almost periodic and periodic solutions of differential equations driven by the fractional Brownian motion with statistical application. Stochastics 93(6):886–906
Neuenkirch A, Tindel S (2014) A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise. Stat Inference Stoch Process 17(1):99–120
Nualart D (2006) The Malliavin calculus and related topics. Springer, Berlin
Saussereau B (2014) Nonparametric inference for fractional diffusions. Bernoulli 20(2):878–918
Tudor CA, Viens F (2007) Statistical aspects of the fractional stochastic calculus. Ann Stat 35(3):1183–1212
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Marie, N. On a calculable Skorokhod’s integral based projection estimator of the drift function in fractional SDE. Stat Inference Stoch Process 27, 391–405 (2024). https://doi.org/10.1007/s11203-024-09306-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11203-024-09306-5