Environment Reconstruction for a Class of Diffusions

Abstract We present an algorithm designed to estimate the so-called environment for diffusions driven by a potential. The technique uses observations from a single path to give a non-parametric estimation of the drift coefficient in a stochastic differential equation. We use an ergodic theorem for diffusions developed by K.Itô and H.P.McKean in a straightforward manner. Furthermore, we develop an estimator to approximate the occupation time of the observed path. Subsequently, we asses the algorithm’s effectiveness through simulations. We generate a path of the diffusion along with its associated environment, and then, by applying our algorithm, we compare the resulting environment with the known a priori one. All programming implementations are conducted using MATLAB.