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

Langevin models are widely 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. (Phys Lett A 271(3):217, 2000). The latter has the distinction of being very effective in the context of large data sets. Due to their $$\delta $$ δ -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. (Phys Rev E 97(1):012113, 2018). However, this procedure requests that the correlation length of the noise component is small compared to that of the measured component. In this work, we propose a direct estimation method without this restriction. This allows one to effectively deal with large data sets from a wide range of examples. We discuss the abilities of the proposed procedure using several synthetic examples. Graphic Abstract