Is there a closed expression for a LTI SDE covariance matrix?

Question

I am studying stochastic differential equations and I still don't have a very firm grasp on all the concepts. My question has to do with the following SDE: $$ \frac{d\textbf x (t)}{dt} = A\textbf x + w(t) $$ where $A$ is a stable matrix (meaning its eigenvalues have negative real part), $\textbf x_0 = \textbf 0$ and $w(t)$ is white noise. There is an expression for the covariance matrix: $$ \mathbb{E}[\mathbf x(t) \cdot \mathbf x(t)^T] = P(t) = e^{At}P_0(e^{At})^T + \int_0^t e^{A\tau}(e^{A\tau})^T d\tau $$ I'd like to know if there is a simple expression for the covariance matrix at the limit of infinite time. There is another approach to this problem which would be to use the expression for the derivative of $P$: $$ \frac{dP(t)}{dt} = AP + PA^T + I = 0 $$ But it doesn't seem straightforward to recover $P$ from this equation. I'm not sure if I'm thinking about the problem in the right way but any help would be appreciated.

Answer 1

As you mentioned, the covariance matrix obeys the Lyapunov differential equation

$$ \dot{P}(t) = AP(t) + P(t)A^T + I,\ P(0)=P_0=\mathbb{E}[x_0x_0^T] $$ and substituting the explicit expression for $P(t)$ shows that it is indeed solution, which turns out to be unique in the present case.

So, as you also mentioned, the stationary covariance matrix, $P_\infty$, is the unique solution to the Lyapunov equation

$$AP_\infty+P_\infty A^T+I=0,$$

which exists provided that $A$ has no eigenvalues on the imaginary axis. However, for the fact that $P(t)\to P_\infty$ as $t\to\infty$ to hold, it is necessary and sufficient that $A$ be Hurwitz stable; i.e. all the eigenvalues of $A$ have negative real part.

In such a case, we have that

$$ P_\infty=\int_0^\infty e^{As}e^{A^Ts}ds, $$

which only holds when $A$is Hurwitz.

Another expression is given by

$$ \mathrm{vec}(P_\infty)=-(A\otimes I+I\otimes A)^{-1}\mathrm{vec}(I) $$

where $\mathrm{vec}(\cdot)$ is the vectorization operator and $\otimes$ denotes teh Kroneck product. Note also that $A\otimes I+I\otimes A$ is invertible if and only if $A$ has no eigenvalues on the imaginary axis.

If you want to learn more about Lyapunov equations, you can have a look at the book "Lyapunov matrix equation in system stability and control" by Gajic and Qureshi.