# Testing the fit of an Ornstein-Uhlenbeck process

I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE:

$$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$

where

• $$\lambda > 0$$ is the mean-reversion coefficient
• $$\mu$$ is the long-term mean
• $$\sigma>0$$ is the variance
• $$W_t$$ is the Wiener process

This paper does the following:

For each pair, we first estimate the parameters for the OU model from empirical price data. Then, we use the estimated parameters to simulate price paths according to the corresponding OU process. Based on these simulated OU paths, we perform another MLE and obtain another set of OU parameters as well as the maximum average log-likelihood $$l$$. As we can see, the two sets of estimation outputs (the rows names “empirical” and “simulated”) are very close, suggesting the empirical price process fits well to the OU model.

They find the parameters via Maximum Likelihood Estimate (MLE), then simulate a fresh OU process with those parameters and find again the parameters of the simulated process. Then they compare the two sets of parameters, and if they are close, they claim the original process fits well to the OU model.

What is the mathematical justification for this?