Relating Ornstein–Uhlenbeck and AR(1) Mean-Reversion Models

Summary

The document compares the continuous-time Ornstein–Uhlenbeck (OU) process with the discrete-time AR(1) model for stationary, mean-reverting data such as a spread. It presents OU as a continuous-time model with an analytic solution and explains that a simple discretization can produce an AR(1)-like update. This connection offers a way to relate model parameters and provides an interpretation of mean-reversion behavior through the OU framework.

The answer gives an Euler-style parameter translation involving the sampling interval, mean-reversion level, and noise scale, and suggests using OU to estimate steady-state parameters. It claims the models are interchangeable when used at the same frequency, but the excerpt does not establish this with derivation or evidence. The displayed discretization is approximate; exact discrete-time sampling of an OU process has different coefficient relationships. No comparison of estimation robustness, precision, or pair-trading results is provided.

Key ideas

• The OU process models mean reversion in continuous time, while AR(1) represents a discrete-time process.
• A simple discretization of OU produces an AR(1)-style update with related parameters.
• The sampling interval is part of the parameter mapping presented in the answer.
• The excerpt suggests OU can help interpret steady-state behavior in stationary data.
• The stated equivalence is not demonstrated empirically, and the displayed discretization is approximate.

Tags

# Ornstein versus AR(1) for modeling stationary data


# Ornstein versus AR(1) for modeling stationary data












I've come across several posts regarding parameter estimation for O-U models given some stationary data (say, some sort of mean reverting spread), but I can't seem to find an answer as to why modeling the data as a continuous O-U bears a benefit over modeling it as an AR(1) process. Are the parameters more robust/precise when treating the process as O-U versus AR(1)? I suppose O-U may give better estimates at higher frequencies. Any insight would be great.

## Answer by user12348 (score 4, accepted)

https://quant.stackexchange.com/a/11056

O-U is continuous time mean reverting process, hence used to model stationary series. It has closed form analytic solution. This allows insight into stationary processes and act like asymptotic limiting case for calculating coefficients that matter.

EDIT: You can see AR(1) below $$x_{k+1} = c + a x_k + b\varepsilon_k$$ and by substituting c=θμΔt, a=−θΔt and $b = \sigma\sqrt{\Delta t} \space$ you will get OU $$ x_{k+1} = \theta(\mu - x_k)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}$$

This is simple discretization to show they are same and how the parameters can be translated. O-U can be used to detect the steady state parameters. As you see paramaters are interchangeable, frequency used in AR and O-U should be same, then it will be frequency agnostic. I am doing some work on pair trading using O-U I will re-edit at some later time.

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