The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there, and let $\Pi$, $K$, and the filter process $m^{\mathrm f}$ be as in The Kalman-Bucy Filter Equation and Its Solution. Write $e_t:=X_t-m^{\mathrm f}_t$ (componentwise) for the estimation error. Then, for every $t\in[0,T]$:

1. (Conditional expectation) For every $i\in\{1,\dots,l\}$, every conditional expectation of $X^{i}_t$ given $\mathcal{G}_t$ is almost surely equal to $(m^{\mathrm f}_t)^{i}$:

2. (Error covariance) With the expectation and covariance, $\mathbb{E}[e^{i}_t]=0$ for every $i$, and

3. (Independence of the error from the observations) The $\sigma$-algebras $\sigma(e^{1}_t,\dots,e^{l}_t)$ and $\mathcal{G}_t$ are independent; in particular $\operatorname{Cov}(e^{i}_t,u^{j}_r)=0$ for all $i,j$ and all $0\le r\le t$.