Consider the stochastic differential equation

$$d Z_t = M Z_t dt + dB_t,$$

where $M in mathbb{R}^{ntimes n}$ is assumed to have eigenvalues that have negative real parts, $B_t$ is $n$-dimensional spherical brownian motion.

The solution of this SDE is given by

$$ Z_t = exp(Mt)Z_0 + int_0^t exp(M(t-s)) dB_s$$

and has the distribution $$Z_t sim mathcal{N}left(exp(Mt)Z_0,2int_0^t exp(M(t-s))exp(M^T(t-s))dsright).$$

Now, consider the function $$f(z) = z^TSigma z.$$

I am trying to understand for what choice of $Sigma$ positive (semi-)definite we have that $mathbb{E}(Z_t^TAZ_t)$ is decreasing.

In the case of a deterministic equation, this yields to the Lyapunov equation, such that all $Sigma$ that solve $$MSigma + Sigma M^T = – Q,$$ for some positive definite $Q$, yield a Lyapunov function for the system

But in the stochastic case, we obtain an additional term that results from the fact that

$$ mathbb{E}(Z_t^TSigma Z_t) = mathbb{E}(Z_t)^TSigma mathbb{E}(Z_t) + operatorname{tr}(Sigma C(t))$$

with $C(t) = operatorname{Cov}(Z_t)$.

Differentiating yields

$$ mathbb{E}(Z_t) Sigma M mathbb{E}(Z_t) + operatorname{tr}(Sigma (M C(t) + C(t)M^T))$$

And then I’m stuck. Theoretically, $C(infty)^{-1}$ should work, but I also don’t know how to show that.

So, any help on how to determine the general form of the solutions would be appreciated!