Let $n,m,r\ge 1$. Let $(\Omega,\mathcal F,\mathbb P)$ support a Gaussian initial state $x_0:\Omega\to\mathbb R^n$, a standard [Brownian motion](/page/Brownian%20Motion) $w:[0,\infty)\times\Omega\to\mathbb R^r$, and a standard Brownian motion $v:[0,\infty)\times\Omega\to\mathbb R^m$, with $x_0$, $w$, and $v$ independent. Let $A:[0,\infty)\to\mathbb R^{n\times n}$, $B:[0,\infty)\to\mathbb R^{n\times m}$, $G:[0,\infty)\to\mathbb R^{n\times r}$, $C:[0,\infty)\to\mathbb R^{m\times n}$, $D:[0,\infty)\to\mathbb R^{m\times m}$, and $u:[0,\infty)\to\mathbb R^m$ be bounded and continuous on each finite time interval. Let $x:[0,\infty)\times\Omega\to\mathbb R^n$ and $y:[0,\infty)\times\Omega\to\mathbb R^m$ solve

\begin{align*} dx(t)=A(t)x(t)\,dt+B(t)u(t)\,dt+G(t)\,dw(t) \end{align*}

with $x(0)=x_0$, and

\begin{align*} dy(t)=C(t)x(t)\,dt+D(t)\,dv(t) \end{align*}

with deterministic initial observation $y(0)=0$. Assume $R(t):=D(t)D(t)^\top$ is positive definite for every $t\ge 0$, and let $P(0):=\operatorname{Cov}(x_0-\mathbb E[x_0])\ge 0$ be finite. Let $\mathcal Y_t$ be the completed sigma-algebra generated by $\{y(s):0\le s\le t\}$. The conditional mean $\hat{x}(t)=\mathbb E[x(t)\mid \mathcal Y_t]$ satisfies, as a semimartingale identity up to indistinguishability,

\begin{align*} d\hat{x}(t) = A(t)\hat{x}(t)\,dt+B(t)u(t)\,dt+K(t)\bigl(dy(t)-C(t)\hat{x}(t)\,dt\bigr). \end{align*}

Here

\begin{align*} K(t) = P(t)C(t)^\top R(t)^{-1}. \end{align*}

The covariance $P(t)=\mathbb E[(x(t)-\hat{x}(t))(x(t)-\hat{x}(t))^\top]$ is locally absolutely continuous and satisfies the Riccati differential equation for $\mathcal L^1$-almost every $t\ge 0$:

\begin{align*} \dot P(t)=A(t)P(t)+P(t)A(t)^\top+G(t)G(t)^\top-P(t)C(t)^\top R(t)^{-1}C(t)P(t). \end{align*}