Non - parametric estimation of the di usion coe cient from noisy data

We consider a diusion process (Xt) t≥0 , with drift b(x) and diusion coecient σ(x). At discrete times t k = kδ for k from 1 to M , we observe noisy data of the sample path, Y kδ = X kδ + ε k. The random variables (ε k) are i.i.d, centred and independent of (Xt). The process (Xt) t≥0 is assumed to be strictly stationary, β-mixing and ergodic. In order to reduce the noise eect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that ∆ = pδ is small whereas M δ is large. Then, the diusion coecient σ 2 is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.