Semiparametric estimation of a functional of the drift coefficient for a non-homogeneous dynamical system with small noise

We consider the problem of semiparametric estimation of functionals of the drift coefficient of a dynamical system by the observation of a non homogeneous diffusion process with small diffusion coefficient. Supposing that the drift coefficient is an unknown (smooth) function we propose a asymptotic minimax lower bound on the risk of all estimators and then we present asymptotically efficient estimators in the sense of this bound. The estimated integral-type functionals are defined with the help of the (unknown) solution of the deterministic limit dynamical system. Some particular and interesting functionals are presented as examples.