Improved nonparametric estimation of the drift in diffusion processes

In this paper, we have considered the robust adaptive nonparametric estimation problem for the drift coefficient in diffusion processes. It has been shown that the initial estimation problem can be reduced to the estimation problem in a discrete time nonparametric heteroscedastic regression model by using the sequential approach. We have developed a new sharp model selection method for estimating the unknown drift function using the improved estimation approach. An adaptive model selection procedure based on the improved weighted least square estimates has been proposed. It has been established that such estimate outperforms in non-asymptotic mean square accuracy the procedure based on the classical weighted least square estimates. Sharp oracle inequalities for the robust risk have been obtained.