Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
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Assuming that the vector valued process $X_t\in\mathbb{R}^n$ can be described via $$ d X_t = -\Psi X_t \,dt + \sigma\, dW_t $$ where $\Psi=(\psi_1,\ldots,\psi_n)$ and $\sigma=(\sigma_1\,\ldots,\sigma_n)$ are both constant and diagonal matrices, then yes.
The $i$th component is $$ dx_{i,t} = -\psi_i x_{i,t}\, dt + \sigma_i\, dW_{i,t} $$ which is a univariate OU process.
Note that if $\sigma$ is not diagonal, then the component processes are still OU, but are no longer independent (as noted by the comment below).
answered Aug 23, 2019 at 3:39
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1$\begingroup$ I think only $\Psi$ needs to be diagonal. If $\sigma$ is not diagonal, the components would still marginally be univariate OU processes, although the components would not be independent. $\endgroup$ Commented Aug 23, 2019 at 7:02
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$\begingroup$ @JarleTufto very true, I've added that to the answer. $\endgroup$ Commented Aug 23, 2019 at 14:18
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