The multivariate Ornstein–Uhlenbeck process is defined as the following \begin{equation} dX(t) = - I_p X(t) dt + \sqrt{2}I_p dW(t) \end{equation} where $I_p$ is an $p \times p$ identity matrix, and $W(t)$ is an $p$-dimensional standard Wiener process. Let $X(0)$ be standard $p$-dimensional Gaussian.

I am looking for the unconditional cross covariance matrix $\text{E}(X(0)X(t))$. Could anyone please help me with this? As of now, I have only found cross-covariance matrix conditioning on $X(0)$, $\text{E}(X(s)X(t)|X(0)=x)$, on Page 16 in "Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein-Uhlenbeck" or in Theorem 2 in "Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process".

I am not sure if unconditional cross covariance follows from these. In Theorem 2, $\text{E}(X(s)X(t)|X(0)=x)$ gets $0$ if one of $s$ or $t$ is $0$. "Mathematical properties" in Ornstein–Uhlenbeck process Wikipedia gives the solution for the one-dimensional case. Thanks a lot!



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