Let $X_t$ be defined by the SDE

$$ dX_t = A(t, X_t)dt + dW_t $$

where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ dependency, the process would be an $n$ dimensional OU process, where the eigenvalues of matrix $A$ in $$ A(X_t) = A\cdot X_t $$ all have negative real part. In this case it can be seen that it would reach an equilibrium (given by an invariant measure), so $X_t$ would exponentially fast approach a stationary solution. But what happens in the case of a $t$-periodic $A(t,X_t)$? Will the process be cyclostationary/what can we say about it?

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    $\begingroup$ I assumed the process is stable (so all the eigenvalues of the matrix $A$ are negative). Is that what you mean? $\endgroup$ – gerd Oct 26 '18 at 13:16

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