  A non-parametric diffusion model with an additive fractional Brownian motion
noise is considered in this work. The drift is a non-parametric function that
will be estimated by two methods. On one hand, we propose a locally linear
estimator based on the local approximation of the drift by a linear function.
On the other hand, a Nadaraya-Watson kernel type estimator is studied. In both
cases, some non-asymptotic results are proposed by means of deviation
probability bound. The consistency property of the estimators are obtained
under a one sided dissipative Lipschitz condition on the drift that insures the
ergodic property for the stochastic differential equation. Our estimators are
first constructed under continuous observations. The drift function is then
estimated with discrete time observations that is of the most importance for
practical applications.
