Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an \reftext{def:admissible-control-2026a}{admissible control} $\alpha$, and the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state and observations} $X^{\alpha}$, $u^{\alpha}$, with notation and fixed versions as in those items. Let $c$, $\gamma$ be the correction processes and $\widehat X$ the \reftext{lem:controlled-state-conditional-expectation-2026a}{controlled estimator}, with the fixed versions of \ref{lem:controlled-state-superposition-2026a} and \ref{lem:controlled-state-conditional-expectation-2026a}, and let $K$ and $m^{\mathrm f}$ be the gain and filter process of \ref{thm:kalman-bucy-filter-solution-2026a}.

Let $t\in(0,T]$, let $k'\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}, and let $f$ assign to each $r\in[0,t]$ a real $k'\times\tilde l$ matrix $f(r)$ with \reftext{def:continuity-closed-interval-c54-2026b}{continuous} entries. Define, for $1\le i\le k'$ (fixed versions),

with the \reftext{def:mean-square-riemann-integral-2026a}{mean-square Riemann integral} (existing by \ref{lem:mean-square-riemann-integral-existence-2026a} and claims 1-2 of \ref{lem:mean-square-riemann-integral-properties-2026a}) and \reftext{thm:vector-wiener-integral-gaussian-2026a}{Wiener integrals} with the versions fixed as in the model; set $\int_0^0 f(r)\,du^{\alpha}_r:=0$, consistently with \ref{lem:observation-stieltjes-adapted-2026a}. Then:

\textbf{1. (Decomposition)} Componentwise and \reftext{def:almost-surely-2026a}{almost surely},

where the first integral on the right is the observation integral of \ref{lem:observation-stieltjes-adapted-2026a} and the last is the mean-square Riemann integral (componentwise, with the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}).

\textbf{2. (Riemann-Stieltjes approximation)} For $n\ge1$ put $x_p=pt/n$ ($0\le p\le n$). Then, componentwise, with $\lVert\cdot\rVert_2$ the mean-square norm of \ref{def:square-integrable-mean-square-2026a},

\textbf{3. (Controlled filter equation)} Componentwise and almost surely, for every $t\in[0,T]$,

with $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$; equivalently, $\widehat X$ is a \reftext{def:linear-sde-mean-square-solution-2026a}{mean-square solution} of the linear stochastic differential equation with coefficient $A-K\tilde E$, forcing family $\bigl(K(r)\tilde E(r)X^{\alpha}_r+B(r)\alpha_r\bigr)_r$, noise matrix $K\tilde\varepsilon$, and constant initial value $\mathbb{E}[\xi]$. Moreover, any family of $l$-tuples of square-integrable random variables whose components are mean-square continuous and which satisfies the displayed equation agrees with $\widehat X$ almost surely at each time.