TITLE: Variance of Multi-Dimensional Ornstein Uhlenbeck process
QUESTION [5 upvotes]: I am trying to compute the asymptotic variance of OU process
$$
d X_t = - H X_t dt + S dW_t
$$
where $X_t$ takes value in $R^d$. $H$ and $S$ are $d\times d$ matrices that does not have $HS = SH$ in general. How to compute its variance at $t$?
Things are classical for one dimensional case as
$$
Var (X_t) = \sigma^2 / \mu (1-e^{-\mu t})
$$

REPLY [4 votes]: the large-$t$ limit $\rho$ of the variance of $X_t$ is given by 
$$\rho=\int_0^\infty \exp(-tH)SS^{\dagger}\exp(-tH^{\dagger})dt$$
where the superscript $\dagger$ indicates the transpose (or the conjugate transpose if the matrix is complex); this result holds also if $S$ is $d\times d'$ with $d'\leq d$; see, for example, page 11 of these notes, which also show how to evaluate the integral by Schur decomposition of $H$.