Proof of Conditional Expectation and Estimation Error of the Extended Controlled State

Fix the approximating sequence chosen in Controlled State of an Extended Admissible Control and let $c^{\alpha}$ be the fixed family from that definition, so that $X^{\alpha,i}_t=X^{i}_t+c^{\alpha,i}_t$ and $\widehat X^{i}_t=m^{\mathrm f,i}_t+c^{\alpha,i}_t$ hold pointwise on $\Omega$ for the fixed versions.

Claim 1. By claims 2 and 3 of The Kalman-Bucy Filter Equation and Its Solution, each component family of the filter process $m^{\mathrm f}$ is mean-square continuous (this is part of the solution property in claim 2) and each $m^{\mathrm f,i}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable (claim 3). By claim 2(a) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control, the same holds for $c^{\alpha}$. Sums of mean-square continuous families are mean-square continuous (triangle inequality, claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm), and a sum of two random variables, each almost surely equal to a $\mathcal{G}_t$-measurable square-integrable one, is almost surely equal to the sum of those versions, which is $\mathcal{G}_t$-measurable and square-integrable. This proves claim 1.

Claim 2. Fix $t$ and $i$. By claim 2(a) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control, choose a $\mathcal{G}_t$-measurable square-integrable $\bar c$ with $c^{\alpha,i}_t=\bar c$ almost surely, and by claim 3 of The Kalman-Bucy Filter Equation and Its Solution choose a $\mathcal{G}_t$-measurable square-integrable $\bar m$ with $m^{\mathrm f,i}_t=\bar m$ almost surely.

First, $\bar m$ is a conditional expectation of $X^{i}_t$ given $\mathcal{G}_t$. Indeed, by Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables some conditional expectation $Y$ of $X^{i}_t$ given $\mathcal{G}_t$ exists, and by claim 1 of The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model it satisfies $Y=m^{\mathrm f,i}_t=\bar m$ almost surely; $\bar m$ is $\mathcal{G}_t$-measurable and square-integrable, and the averaging identity in condition (iii) of Conditional Expectation of a Square-Integrable Random Variable is unchanged when $Y$ is replaced by the almost surely equal $\bar m$, so $\bar m$ satisfies all three conditions of that definition.

Second, $\bar c$ is a conditional expectation of $c^{\alpha,i}_t$ given $\mathcal{G}_t$: by the immediate-consequences paragraph of Conditional Expectation of a Square-Integrable Random Variable, a $\mathcal{G}_t$-measurable square-integrable random variable is a conditional expectation of itself, and replacing the conditioned variable $\bar c$ by the almost surely equal $c^{\alpha,i}_t$ leaves the expectations in condition (iii) unchanged.

By linearity (claim 1 of Basic Properties of Conditional Expectation for Square-Integrable Random Variables), $\bar m+\bar c$ is a conditional expectation of $X^{i}_t+c^{\alpha,i}_t=X^{\alpha,i}_t$ given $\mathcal{G}_t$. Now let $Y'$ be any conditional expectation of $X^{\alpha,i}_t$ given $\mathcal{G}_t$; both $Y'$ and $\bar m+\bar c$ are $\mathcal{G}_t$-measurable and square-integrable, so the uniqueness clause of Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables gives $Y'=\bar m+\bar c$ almost surely, and $\bar m+\bar c=m^{\mathrm f,i}_t+c^{\alpha,i}_t=\widehat X^{i}_t$ almost surely. This proves claim 2.

Claim 3. For the fixed versions, pointwise on $\Omega$,

so the identity holds surely, in particular almost surely. The stated consequences are exactly claims 2-3 of The Kalman-Bucy Filter Computes the Conditional Expectation in the Linear-Gaussian Model applied to $e$, as recorded in the statement.

Claim 4. If $\alpha$ is admissible, then by claim 2(c) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control the family $c^{\alpha}$ agrees almost surely at every $t$ with the correction process $c$ of Superposition Decomposition of the Controlled State and Observations for $\alpha$, so $\widehat X_t=m^{\mathrm f}_t+c^{\alpha}_t$ agrees almost surely, componentwise, with the controlled estimator $m^{\mathrm f}_t+c_t$ defined there.

For the convergence statement, let $\bigl((\alpha^{(n)}),D\bigr)$ be an arbitrary approximating sequence of $\alpha$, not necessarily the one chosen in Controlled State of an Extended Admissible Control. By claim 2 of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control applied to this sequence, there is a family $c'$ satisfying claims 2(a) and 2(b) for it, and by claim 2(c) (which covers arbitrary pairs of approximating sequences), $c'^{\,i}_t=c^{\alpha,i}_t$ almost surely for every $t$ and $i$; almost sure equality leaves mean-square norms unchanged, so the conclusion of claim 2(b) for $c'$ holds verbatim with $c^{\alpha}$ in place of $c'$. The controlled estimator of $\alpha^{(n)}$ is $\widehat X_t(\alpha^{(n)})=m^{\mathrm f}_t+c^{(n)}_t$ with $c^{(n)}$ the correction process of $\alpha^{(n)}$, so componentwise and pointwise $\widehat X^{i}_t(\alpha^{(n)})-\widehat X^{i}_t=c^{(n),i}_t-c^{\alpha,i}_t$, and the assertion — continuity in $t$ of the summed mean-square norms, existence of the maxima by Extreme Value Theorem on a Compact Interval, and convergence of the maxima to $0$ — is precisely claim 2(b) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control for this sequence, transferred to $c^{\alpha}$ as just explained. $\square$