# Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

## TL;DR

This paper introduces a direct method to estimate parameters of a non-Markovian stochastic process driven by Ornstein-Uhlenbeck noise, using moments of increments and stochastic Taylor-expansion, simplifying analysis from time series data.

## Contribution

It presents a novel, moment-based estimation approach for non-Markovian processes driven by Ornstein-Uhlenbeck noise, avoiding complex Bayesian or embedding techniques.

## Key findings

- Derivation of analytic expressions for moments of process increments.
- Establishment of a regression-based parameter estimation method.
- Potential for improved analysis of non-Markovian stochastic processes.

## Abstract

A scalar Langevin-type process $X(t)$ that is driven by Ornstein-Uhlenbeck noise $\eta(t)$ is non-Markovian. However, the joint dynamics of $X$ and $\eta$ 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 dynamic of $X$ and its temporal derivative $\dot X$ is analyzed. Subsequently the results can be used to determine the process parameters of $X$ and $\eta$. 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 $\tau$ 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.