Nonparametric estimation with small-noise diffusion processes

We consider a generalization of a nonparametric estimation problem for an integral-type functional by considering Frechet differentiable functionals on Hilbert spaces with observations from nondegenerate diffusion-coefficient stochastic differential equations. The method of estimation is based on obtaining a minimax lower bound on the risk functions of all possible nonparametric estimators and then constructing an asymptotically efficient estimator in the sense of this bound when the functionals satisfy certain conditions. Then we show, by examples, how this general approach can be used to solve certain types of nonparametric estimation problems.