Parameter estimation by contrast minimization for noisy observations of a diffusion process

We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times i delta, i=1, horizontal ellipsis , N. Assuming that the sampling interval tends to 0 while the total length-time interval tends to infinity, we prove limit theorems for functionals associated with the observations, based on local means of the sample. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent. Some examples are discussed with numerical simulations.