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

We consider N independent stochastic processes (Xᵢ(t),t∈ [0,T]), i=1,… ,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 R. Projection estimators are defined on finite dimensional subsets of L²(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.