Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

A scalar Langevin-type process X(t) that is driven by Ornstein-Uhlenbeck noise η(t) is non-Markovian. However, the joint dynamics of X and η is described by a Markov process in two dimensions. But even though there exists a variety of techniques for the analysis of Markov processes, it is still a challenge to estimate the process parameters solely based on a given time series of X. Such a partially observed 2D process could, e.g., be analyzed in a Bayesian framework using Markov chain Monte Carlo methods. Alternatively, an embedding strategy can be applied, where first the joint dynamics of X and its temporal derivative X[over ̇] is analyzed. Subsequently, the results can be used to determine the process parameters of X and η. In this paper, we propose a more direct approach that is purely based on the moments of the increments of X, which can be estimated for different time-increments τ from a given time series. From a stochastic Taylor expansion of X, analytic expressions for these moments can be derived, which can be used to estimate the process parameters by a regression strategy.