Conditional Expectation and Estimation Error of the Controlled State

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an admissible control $\alpha$, and the controlled state $X^{\alpha}$, with notation and fixed versions as in those items. Let $m^{\mathrm f}$ and $\Pi$ be the filter process and covariance assignment of The Kalman-Bucy Filter Equation and Its Solution, let $e_t:=X_t-m^{\mathrm f}_t$ be the estimation error of The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model, and let $c$ be the correction process of Superposition Decomposition of the Controlled State and Observations. Define the controlled estimator

$$\widehat X_t:=m^{\mathrm f}_t+c_t\qquad(0\le t\le T),$$

componentwise, with the fixed versions of those items. Then, for every $t\in[0,T]$:

1. (Regularity and adaptedness) Each component family $(\widehat X^{i}_t)_{t\in[0,T]}$ is mean-square continuous, and each $\widehat X^{i}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable.

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

3. (Error invariance) For every $i$: $X^{\alpha,i}_t-\widehat X^{i}_t=e^{i}_t$ almost surely. Consequently, by claims 2-3 of The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model, with the expectation and covariance: $\mathbb{E}[e^{i}_t]=0$ for every $i$, the matrix $\bigl(\operatorname{Cov}(e^{i}_t,e^{j}_t)\bigr)_{1\le i,j\le l}$ equals $\Pi(t)$, and the $\sigma$-algebras $\sigma(e^{1}_t,\dots,e^{l}_t)$ and $\mathcal{G}_t$ are independent.

4. (Orthogonality to observation-determined variables) For every $i$ and every square-integrable random variable $\zeta$ that is almost surely equal to a $\mathcal{G}_t$-measurable random variable: $e^{i}_t\,\zeta$ is integrable and $\mathbb{E}[e^{i}_t\,\zeta]=0$.