I am looking for the solution/distribution of the multivariate Ornstein-Uhlenbeck (mOUP) in the following form, $$ dX_t = AX_t + B\; dt + \Sigma dW_t $$ where $A,\Sigma$ are $N\times N$ matrices and $B$ is a vector. Whilst the solution for the mOUP in the following form, $$ dX_t = \Theta(X_t-\mu)\;dt + \Sigma dW_t $$ is available online, I cannot find it in my desired form. Furthermore, if the solution includes an Ito integral, I would appreciate its covariance structure as I am trying to exactly numerically simulate the mOUP at discrete time-points.

Thank you !


1 Answer 1


So I think I found a simple way to go from the other form of the OUP to this one and therefore to use the solution from the other OUP.

Given the SDE, $$ dX_t = \Theta(X_t-\mu)\;dt + \Sigma dW_t, $$ The solution is normally distributed with mean vector, $$ M(t) = e^{-\Theta t}X_0 + (I-e^{-\Theta t})\mu $$ and covariance, $$ C(t) = \int_0^te^{\Theta (s-t)}\Sigma \Sigma^{\top}e^{\Theta^{\top} (s-t)}\;ds $$ And therefore we can take $A=-\Theta$ and $-A^{-1}B=\mu$ which transforms the solution to be normally distributed with mean vector, $$ M(t) = e^{A t}X_0 - (I-e^{A t})A^{-1}B $$ and covariance, $$ C(t) = \int_0^te^{A (t-s)}\Sigma \Sigma^{\top}e^{A^{\top} (t-s)}\;ds $$ And so we have the result in terms of $A,B$.

The reference I used for the solution in the case of the other form is:

Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process P. Vatiwutipong & N. Phewchean, Advances in continuous and discrete models.

  • 1
    $\begingroup$ It's good. I'm writing the answer but you finally found it by your own. It suffices to apply the same technique but for multivariate vector. $\endgroup$
    – NN2
    Commented May 13, 2023 at 15:54

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