Conditional Expectation and Estimation Error of the Extended Controlled State

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an extended admissible control $\alpha$ with values in $\mathbb{R}^{k}$, and its controlled state $X^{\alpha}=X+c^{\alpha}$, 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, and 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. Define the extended controlled estimator

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

componentwise, with the fixed versions of those items. Then:

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

2. (Conditional expectation) For every $t\in[0,T]$ and 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 $t\in[0,T]$ and 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. (Consistency and convergence) If $\alpha$ is an admissible control, then for every $t\in[0,T]$ the tuple $\widehat X_t$ agrees almost surely, componentwise, with the controlled estimator already defined for admissible controls. Moreover, for any approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ of $\alpha$, writing $\widehat X(\alpha^{(n)})$ for the controlled estimator of $\alpha^{(n)}$ from Conditional Expectation and Estimation Error of the Controlled State: the maxima over $t\in[0,T]$ of the continuous functions $t\mapsto\sum_{i=1}^{l}\lVert\widehat X^{i}_t(\alpha^{(n)})-\widehat X^{i}_t\rVert_{2}$ tend to $0$ as $n\to\infty$, with the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product and the maxima existing by Extreme Value Theorem on a Closed Real Interval.