Efficient big data analysis for ergodic diffusion models on the basis of discrete data.

We consider a big data analysis problem for diffusion processes in the framework of nonparametric estimation for ergodic diffusion processes based on observations at discrete time moments in the case when diffusion coefficients are unknown. To this end we use the model selection method developed by Galtchouk and Pergamenshchikov (2019). In this paper through the oracle inequalities obtained by Galtchouk and Pergamenshchikov (2019) we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the regularity of the drift coefficient is unknown. To this end, for the first time for such problem, we found in the explicit form the celebrated Pinsker constant which is the sharp lower bound for the minimax squared accuracy normalized by the optimal convergence rate, i.e. it provides the best potentially possible estimation accuracy. Finally, we show that the accuracy of the model selection procedure asymptotically coincides with this lower bound, i.e. the constructed procedure is efficient