Non-parametric estimation of a Langevin model driven by correlated noise

TL;DR

This paper introduces a new non-parametric estimation method for non-Markovian Langevin models driven by correlated noise, enabling effective analysis of large datasets without previous restrictions on noise correlation length.

Contribution

A novel version of the direct estimation method is proposed, allowing for the estimation of non-Markovian Langevin models with correlated noise without the small correlation length restriction.

Findings

Effective estimation on synthetic datasets demonstrated.

Method handles larger data sets and wider correlation ranges.

Improves upon previous limitations in non-Markovian Langevin modeling.

Abstract

Langevin models are frequently used to model various stochastic processes in different fields of natural and social sciences. They are adapted to measured data by estimation techniques such as maximum likelihood estimation, Markov chain Monte Carlo methods, or the non-parametric direct estimation method introduced by Friedrich et al. The latter has the distinction of being very effective in the context of large data sets. Due to their $\delta$-correlated noise, standard Langevin models are limited to Markovian dynamics. A non-Markovian Langevin model can be formulated by introducing a hidden component that realizes correlated noise. For the estimation of such a partially observed diffusion a different version of the direct estimation method was introduced by Lehle et al. However, this procedure includes the limitation that the correlation length of the noise component is small compared to that of the measured component. In this work we propose another version of the direct estimation method that does not include this restriction. Via this method it is possible to deal with large data sets of a wider range of examples in an effective way. We discuss the abilities of the proposed procedure using several synthetic examples.