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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 $$ Var (X_t) = \sigma^2 / \mu (1-e^{-\mu t}) $$

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  • $\begingroup$ Are $H$ and $S$ constants? I haven't tried, but would expect most of the one-dimensional proofs to go through without trouble. For instance, Mehler's formula is available in higher dimensions. $\endgroup$ Mar 30, 2017 at 23:48
  • $\begingroup$ Also, what initial conditions do you have in mind for $X_0$? $\endgroup$ Mar 30, 2017 at 23:52
  • $\begingroup$ Thanks. $H$ and $S$ are $d\times d$ matrices that does not have $HS = SH$ in general (I added that into my question). For initial vector it does not matter since the stationary distribution here is not dependent on $X_0$. $\endgroup$ Mar 31, 2017 at 7:12

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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 $H$.

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