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

TL;DR

This paper introduces a variational approach to nonparametrically estimate the drift function of stochastic differential equations from empirical data, utilizing kernel regularization and the Fokker-Planck equation.

Contribution

It proposes a novel variational formulation and a closed-form solution for drift estimation from empirical densities, applicable to Langevin-type equations and adaptable to other noise models.

Findings

Effective drift estimation demonstrated on Langevin equations

Method achieves closed-form solutions for the variational problem

Generalizable to different noise models

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.