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Consider a diffusion process ( X   t  )  t  ≥ 0 , with unknown drift  b ( x ) and diffusion coefficient  σ ( x ), which is strictly stationary, ergodic and  β -mixing. At discrete times  t   k   =  k  δ  for  k  from 1 to  N , we have at disposal noisy data of the sample path,  Y   k  δ   =  X   k  δ  + ε   k  . The random variables   ( ε   k  ) are i.i.d., centred and independent of ( X   t  ). In order to reduce the noise effect, we split data into groups of equal size  p  and build empirical means. The group size  p  is chosen such that Δ =  p  δ  is small whereas  N  δ  is large. Then, the drift function  b  is estimated in a compact set  A  in a non-parametric way using a penalized least squares approach. We obtain a bound for the risk of the resulting adaptive estimator. Examples of diffusions satisfying our assumptions are given and numerical simulation results illustrate the theoretical properties of our estimators.

Non-parametric drift estimation for diffusions from noisy data