The Limiting Cost Along the Approximate Kalman Policy Is the Optimal Value of the Fluctuation LQG Problem

Adopt the setting, hypotheses \textbf{(H1)}--\textbf{(H4)}, and notation of the \reftext{lem:approximate-kalman-policy-2026a}{approximate Kalman filter and policy lemma} for the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data} of the \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple} $(S,A,P)$: in particular the \reftext{def:natural-numbers-2026a}{natural numbers} $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the horizon $T>0$, the matrices $\mathcal{E}_t$, $\mathcal{B}_t$, $\tilde{\mathcal{E}}_t$, $\Theta^\star_t$, $\tilde{\Theta}^\star_t$ of the fluctuation LQG data, the symmetrized coefficient matrices $Q_t$, $V_t$, $R_t$, $\hat{F}$, the Riccati family $Z=(Z_t)_{t\in[0,T]}$ of (H2) with $W_t=Z_t\mathcal{B}_t+\tfrac12 V_t$, the matrix $\Pi_0$ of (H4), and the filter covariance $\Pi=(\Pi_t)_{t\in[0,T]}$ of conclusion 2 of the policy lemma. For real matrices $A'$ and $B'$ with $l$ rows and $l$ columns write $A'\cdot B'=\sum_{\gamma,\delta=1}^{l}A'^{\gamma\delta}B'^{\gamma\delta}$, as in \reftext{prop:kalman-policy-cost-limit-2026a}{the cost-limit proposition}.

Consider moreover a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$ whose state dimension and observation dimension (the numbers $l$ and $\tilde{l}$ of the model definition) are the present $l$ and $\tilde{l}$, and whose Brownian dimension (the number $m$ of the model definition) is a natural number $m^\circ\ge1$, the letter $m$ remaining the control dimension of the fluctuation setting; write $(A^\circ,\varepsilon^\circ,\tilde{E}^\circ,\tilde{\varepsilon}^\circ,\xi,W^\circ)$ for its data, written $(A,\varepsilon,\tilde{E},\tilde{\varepsilon},\xi,W)$ in the model definition, the letters $A$ and $W$ being otherwise engaged here and the letters $\varepsilon$ and $\tilde{\varepsilon}$ being freed for the notation of \reftext{prop:kalman-policy-cost-limit-2026a}{the cost-limit proposition}, whose filter error is written $\varepsilon^N$. Assume, with the \reftext{def:product-real-matrices-2026a}{matrix product} and \reftext{def:transpose-real-matrix-2026a}{transpose}:

\textbf{(i)} $A^\circ(t)=\mathcal{E}_t$ and $\tilde{E}^\circ(t)=\tilde{\mathcal{E}}_t$ for every $t\in[0,T]$;

\textbf{(ii)} $\varepsilon^\circ(t)\,\varepsilon^\circ(t)^{\top}=\Theta^\star_t$ and $\tilde{\varepsilon}^\circ(t)\,\tilde{\varepsilon}^\circ(t)^{\top}=\tilde{\Theta}^\star_t$ for every $t\in[0,T]$ --- so the matrices $\Theta(t)=\varepsilon^\circ(t)\varepsilon^\circ(t)^{\top}$ and $\tilde{\Theta}(t)=\tilde{\varepsilon}^\circ(t)\tilde{\varepsilon}^\circ(t)^{\top}$ of the model definition equal $\Theta^\star_t$ and $\tilde{\Theta}^\star_t$;

\textbf{(iii)} $\mathbb{E}[\xi^{\gamma}]=0$ and $\operatorname{Cov}(\xi^{\gamma},\xi^{\delta})=\Pi_0^{\gamma\delta}$ for all $\gamma,\delta\in\{1,\dots,l\}$, with the \reftext{def:expectation-variance-2026a}{expectation} and the \reftext{def:covariance-square-integrable-2026a}{covariance}.

Take the control dimension $k$ of the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled system} to be $m$, and the control matrix assignment to be $B(t)=\mathcal{B}_t$ ($0\le t\le T$) --- an assignment of real matrices with $l$ rows and $m$ columns whose entries are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} by conclusion 1 of the policy lemma. Take the \reftext{def:lqg-cost-functional-2026a}{cost data} to be

--- these are cost data with every $R^\circ(t)$ \reftext{def:positive-semidefinite-matrix-2026a}{positive definite}, by conclusion 1 of the policy lemma and the entry formulas recorded there. Write $J[\alpha]$ for the \reftext{def:extended-lqg-cost-2026a}{linear-quadratic-Gaussian cost} of each \reftext{def:extended-admissible-control-2026a}{extended admissible control} $\alpha$ with values in $\mathbb{R}^{m}$ for these data. Then:

\textbf{1. (Identification of the LQG ingredients.)} The family $Z$ is a symmetric continuous solution of the backward Riccati equation of \reftext{thm:lqg-completion-of-squares-2026a}{the completion-of-squares theorem} for these data, with $Z_t\,B(t)+V^\circ(t)=W_t$ for every $t\in[0,T]$; and the initial covariance matrix $P^{\mathrm{KB}}_0$ (written $P_0$ in that theorem) and the covariance assignment of claim 1 of \reftext{thm:kalman-bucy-filter-solution-2026a}{the Kalman--Bucy filter theorem}, formed for this model, satisfy: $P^{\mathrm{KB}}_0=\Pi_0$, and the covariance assignment assigns to each $t\in[0,T]$ exactly the matrix $\Pi_t$ of the policy lemma.

\textbf{2. (The minimal cost is the limiting cost.)} The optimal value $V^{*}$ of \reftext{thm:lqg-separation-theorem-2026a}{the separation theorem}, formed for these data and the solution $Z$ of conclusion 1, is the minimum of $J$ over all extended admissible controls with values in $\mathbb{R}^{m}$, by \reftext{thm:lqg-separation-extended-2026a}{the separation theorem over extended admissible controls}; it is attained by the closed-loop feedback control of \reftext{lem:closed-loop-feedback-control-2026a}{the closed-loop feedback lemma}; and

the integral being the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,T]$ of an integrand that is continuous on $[0,T]$ --- all entries of $t\mapsto Z_t$ are continuous (part of \textbf{(H2)}), all entries of $t\mapsto\Theta^\star_t$, $t\mapsto R_t^{-1}$, and $t\mapsto\Pi_t$ are continuous by conclusions 1 and 2 of the \reftext{lem:approximate-kalman-policy-2026a}{policy lemma}, and the entries of $t\mapsto W_t=Z_t\mathcal{B}_t+\tfrac12V_t$ and the integrand itself are built from these and the continuous entries of $t\mapsto\mathcal{B}_t$ and $t\mapsto V_t$ (conclusion 1 again) by \reftext{thm:sum-product-continuous-real-2026a}{sums and products of continuous functions} --- and whose Lebesgue and Riemann integrals \reftext{lem:riemann-lebesgue-integral-agree-2026a}{agree}; the right-hand side above is precisely the right-hand side of the limit identity of the cost-limit proposition. In particular, if moreover for each natural number $N\ge1$ a driving system and a projected solution along the approximate Kalman policy $h^N$ are fixed as in \reftext{prop:kalman-policy-cost-limit-2026a}{the cost-limit proposition}, with $J^N[h^N]$, $J^{MF}$, and $\zeta_N$ as there, and the initial-condition hypotheses (I1)--(I2) there hold, then

\textbf{3. (Realizability of the coefficients.)} For the Brownian dimension $m^\circ=l\cdot l+\tilde{l}$ there exist assignments $\varepsilon^\circ$ and $\tilde{\varepsilon}^\circ$ of real matrices with $l$ rows and $m^\circ$ columns, respectively $\tilde{l}$ rows and $m^\circ$ columns, to the points of $[0,T]$, all of whose entries are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$, such that for every $t\in[0,T]$

the last matrix being the zero matrix with $l$ rows and $\tilde{l}$ columns. Such assignments satisfy hypothesis (ii) and conditions (i) and (ii) of the \reftext{def:linear-gaussian-state-observation-model-2026a}{model definition} (every $\tilde{\Theta}^\star_t$ being symmetric positive definite by conclusion 1 of the policy lemma), so hypotheses (i)--(iii) concern a nonempty class of coefficient data; the model's remaining stochastic data --- the Brownian motion and the jointly Gaussian initial vector $\xi$ --- stay hypothesized exactly as in the model definition, as everywhere in this chain.