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

Abstract

By Fabienne Comte∗, Valentine Genon-Catalot∗ Université de Paris, MAP5, CNRS, F-75006, France ∗ We considerN independent stochastic processes (Xi(t), t ∈ [0, T ]), i = 1, . . . , N , de ned by a one-dimensional stochastic di erential 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 de ned on nite dimensional subsets of L(A, dx). We stress that the set A may be compact or not and the di usion coe cient 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.