A ridge estimator of the drift from discrete repeated observations of the solution of a stochastic differential equation

Abstract

This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval $[0,\mathit{T}]$. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.