Autoregression Approximation of a Nonparametric Diiusion Model

We consider a model of small diiusion type where the function which governs the drift term varies in a nonparametric set. We investigate discrete versions of this continuous model with respect to statistical equivalence, in the sense of the asymptotic theory of experiments. It is shown that an Euler diierence scheme as a discrete version of the stochastic diierential equation is asymptotically equivalent in the sense of Le Cam's deeciency distance, when the discretization step decreases with the noise intensity. We thus obtain a nonparametric version of diiusion limit results for autoregression. It follows that in the continuous diiu-sion model, discrete sampling on a uniform grid is asymptotically suucient. The key technical step utilizes the notion of Hellinger process from semimartingale theory.