Adaptive methods for non-parametric estimation of the coefficients of a diffusion.

Summary We study the problem of non-parametric estimation of the coefficients of a one-dimensional diffusion for a discrete observation of the trajectory in the framework of the minimax theory. Two asymptotics are mainly considered: diffusions observed over a fixed time interval (the diffusion coefficient is then estimated, whether it depends on time or space) and stationary diffusions over a time interval that increases with the number of observations (the drift coefficient and the diffusion coefficient are estimated simultaneously). The minimax velocities are calculated when the unknown parameter is subject to a besov constraint. The method is based on an approximation of the diffusion models by regression schemes, and allows the implementation of the wavelet coefficient thresholding techniques used by donoho, johnstone, kerkyacharian and picard for density or regression models.