On the functional estimation of multivariate diffusion processes

We propose a fully nonparametric estimation theory for the drift vector and the diffusion matrix of multivariate diffusion processes. The estimators are sample analogues to infinitesimal conditional expectations constructed as Nadaraya-Watson kernel averages. Minimal assumptions are imposed on the statistical properties of the multivariate system to obtain limiting results. Harris recurrence is all that we require to show strong consistency and asymptotic (mixed) normality of the functional estimates. Hence, the estimation method and asymptotic theory apply to both stationary and nonstationary multivariate diffusion processes of the recurrent type.