Test for OU-Process

Question

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?

So far, everything I've seen is hand-wavy...

Answer 1

$\newcommand\th{\theta}$ $\newcommand\Si{\Sigma}$ The Ornstein--Uhlenbeck (OU) process is a Gaussian process, with the mean and covariance functions being known functions depending on $s\le4$ unknown real-valued parameters of the OU process. So, for the sample $X:=(x_1,\dots,x_n)$, we know the functions $\mu$ and $\Si$ given by $\mu(\th):=E_\th X$ and $\Si(\th):=Cov_\th X$ for all $s$-tuples $\th$ of the real-valued parameters. Then $$Z(\th):=\Si(\th)^{-1/2}(X-\mu(\th))$$ has the standard normal distribution in $\mathbb R^n$.

Now using appropriate estimates of $\th$ and partitioning $\mathbb R^n$ into some number $k>s+1$ of disjoint Lebesgue-measurable sets of nonzero (say the same) Gaussian measure, we can use e.g. a chi-square goodness-of-fit test, with a test statistic distributed approximately as $\chi^2$ with $k-s-1$ degrees of freedom, as described e.g. by Watson.

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Another goodness-of-fit test, based on a strict negative definiteness of Euclidean distance, is described in Section 3 of the paper by Székely and Rizzo.