## Nonparametric sequential estimation of the drift in diffusion processes via model selection.(English)Zbl 1129.62073

Summary: The paper is devoted to nonparametric estimation in the $${\mathcal L}_2$$-metric of the drift coefficient depending on the state variable in ergodic diffusion processes. We make use of sequential kernel estimators and apply a model selection approach. In the non-asymptotic setting, an upper bound for the risk is obtained. In the asymptotic setting, the upper and lower bounds for the risk are given, which provide minimaxity of penalized estimators. In the latter case the proposed procedure is adaptive when the smoothness of the drift is unknown.

### MSC:

 62M05 Markov processes: estimation; hidden Markov models 62G05 Nonparametric estimation 62L12 Sequential estimation