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

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 \({\widehat{b}}\) 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.