Variational estimation of the drift for stochastic differential equations from the empirical density

Abstract

We present a method for the nonparametric estimation of the drift function of certain types of stochastic differential equations from the empirical density. It is based on a variational formulation of the Fokker-Planck equation. The minimization of an empirical estimate of the variational functional using kernel based regularization can be performed in closed form. We demonstrate the performance of the method on second order, Langevin-type equations and show how the method can be generalized to other noise models.