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 , dened by a one-dimensional stochastic dierential equation which are continuously observed throughout a time interval [0, T ] where T is xed. We study nonparametric estimation of the drift function on a given subset A of R. Projection estimators are dened on nite dimensional subsets of L 2 (A, dx). We stress that the set A may be compact or not and the diusion coecient 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. March 28, 201