Calibrating Correlated Ornstein–Uhlenbeck Processes from Residuals

Summary

The document asks for references and a maximum-likelihood method to estimate parameters for two correlated Ornstein–Uhlenbeck processes from observed data. The question raises using a multivariate likelihood and the mean and variance properties of an OU process, but the answer does not provide a joint likelihood derivation or a worked implementation.

Instead, the response recommends calibrating each process separately, then estimating their correlation from residual increments after applying the fitted parameters. This outlines a practical two-stage route for characterizing dependence, but does not explain how it compares with joint maximum likelihood or address sampling frequency, model diagnostics, or uncertainty in estimates. No empirical example or supporting citation is supplied in the answer, so readers seeking a rigorous joint estimator will need additional sources.

Key ideas

• The question concerns maximum-likelihood estimation for two correlated OU processes.
• The answer recommends fitting each process separately before estimating dependence.
• Correlation is then computed from residual increments using the fitted parameters.
• The response does not derive a joint likelihood or provide an implementation.

Tags

# How to estimate parameters for 2 correlated Ornstein-Uhlenbeck processes with maximum likelihood?


# How to estimate parameters for 2 correlated Ornstein-Uhlenbeck processes with maximum likelihood?












I would like to use maximum likelihood to estimate the parameters of two correlated Ornstein-Uhlenbeck processes from empirical data.

Do you have any good references for this? If you have any hints as to how to code it in Matlab, that would also be great.

I suppose I can just use the log-likelihood function for multivariate processes and then I need mean and variance of an Ornstein-Uhlenbeck process, e.g. as described in this answer. Right?

However, even once I have coded it in Matlab, I would still have to reference where I got the formulas from, and there it would be helpful to have some (academic) papers.

I am familiar with the paper by Schwartz and Smith 2000 (Short-term variations and long-term dynamics in commodity prices, Management Science), but this one is on 1 O-U and 1 Brownian Motion.

## Answer by M. Jeunesse (score -1)

https://quant.stackexchange.com/a/25293

I don't know what exactly you want but have a look at the article Calibrating the Ornstein-Uhlenbeck (Vasicek) model.

You calibrate the first one in stand alone, then the second one in stand alone, finally you can compute correlation on the residuals of the increments knowing your parameters.

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