Nonparametric Estimation for I.I.D. Paths of Fractional SDE

Fabienne Comte (1) , Nicolas Marie (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 MODAL'X - Modélisation aléatoire de Paris X

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 $\widehat b$ 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 $\widehat b$. Another bound is directly established on an estimator of $b'$ for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the $\mathbb R$-supported Hermite basis.

Mots clés: Fractional Brownian motion; Nonparametric projection estimator; Stochastic differential equation

Domaines: Statistiques [stat]

Fabienne Comte, Nicolas Marie. Nonparametric Estimation for I.I.D. Paths of Fractional SDE. Statistical Inference for Stochastic Processes, 2021, 24 (3), pp.669-705. ⟨10.1007/s11203-021-09246-4⟩. ⟨hal-02532339v2⟩