# Exact solution and distribution of multivariate Ornstein-Uhlenbeck equation

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 !

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 $$A,B$$.