Nonparametric drift estimation for i.i.d. paths of stochastic differential equations

Abstract

We consider $N$ independent stochastic processes $(X_{i}(t),t\in [0,T])$, $i=1,\ldots ,N$, defined by a one-dimensional stochastic differential equation, which are continuously observed throughout a time interval $[0,T]$ where $T$ is fixed. We study nonparametric estimation of the drift function on a given subset $A$ of ${\mathbb{R}}$. Projection estimators are defined on finite dimensional subsets of ${\mathbb{L}}^{2}(A,dx)$. We stress that the set $A$ may be compact or not and the diffusion coefficient may be bounded or not. A data-driven procedure to select the dimension of the projection space is proposed where the dimension is chosen within a random collection of models. Upper bounds of risks are obtained, the assumptions are discussed and simulation experiments are reported.