Existence of Asymptotically Optimal-Value Observation-Driven Policies

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-2026b}{stationary mean-field triple} $(S,A,P)$, whose stationary co-state $P$ has value $P_0=(P^1_0,\dots,P^l_0)$ at time $0$: in particular the \reftext{def:natural-numbers-2026a}{natural numbers} $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ and \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with their extensions, the \reftext{def:c2-population-cost-extension-2026b}{cost extension} of the \reftext{def:population-cost-data-2026a}{population cost data} $(L,G)$, 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 $Q_t$, $V_t$, $R_t$, $\hat{F}$ --- each $R_t$ symmetric positive definite with continuous inverse by conclusion 1 of the policy lemma --- the Riccati family $Z=(Z_t)_{t\in[0,T]}$ of (H2), fixed throughout, with $W_t=Z_t\mathcal{B}_t+\tfrac12V_t$, the matrix $\Pi_0$ of (H4), and the filter covariance $\Pi=(\Pi_t)_{t\in[0,T]}$ of conclusion 2 of the policy lemma. All entries of the maps $t\mapsto Z_t$ (part of (H2)), $t\mapsto\mathcal{B}_t$, $t\mapsto V_t$, $t\mapsto\Theta^\star_t$, and $t\mapsto R_t^{-1}$ (conclusion 1 of the policy lemma), and $t\mapsto\Pi_t$ (conclusion 2 there) are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$; hence, with the matrix pairing $A'\cdot B'=\sum_{\gamma,\delta=1}^{l}A'^{\gamma\delta}B'^{\gamma\delta}$ of \reftext{prop:kalman-policy-cost-limit-2026a}{the cost-limit proposition}, the map $t\mapsto Z_t\cdot\Theta^\star_t+(W_tR_t^{-1}W_t^{\top})\cdot\Pi_t$ is continuous on $[0,T]$ by \reftext{thm:sum-product-continuous-real-2026a}{sums and products of continuous functions}. Define the real number

the integral being the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,T]$ of this continuous integrand, which \reftext{lem:riemann-lebesgue-integral-agree-2026a}{agrees} with its Riemann integral. Then:

\textbf{1. (Existence of the policies.)} For each natural number $N\ge1$ the approximate Kalman policy $h^N$ of conclusion 3 of the policy lemma is an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} with horizon $T$, control dimension $m$, and $\tilde{l}$ channels; and for every \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} with $l$ states and $\tilde{l}$ observation channels there is a projected \reftext{def:n-agent-controlled-dynamics-2026a}{solution of the controlled $N$-agent dynamics} on $[0,T]$ for $\beta$, $\tilde{\beta}$, that driving system, and $h^N$, as in conclusion 4(a) of the policy lemma --- the state processes, observation processes, and regular event of a solution for the modified family $\beta^{\sharp}$ and policy $h^{\sharp}$ of conclusion 3 there, with the control process replaced by the vector of its first $m$ components --- and every solution for $\beta$, $\tilde{\beta}$, that driving system, and $h^N$ is indistinguishable from it in the sense of the \reftext{thm:n-agent-dynamics-existence-2026a}{uniqueness theorem}.

\textbf{2. (Asymptotic value.)} Suppose that for each $N\ge1$ such a driving system and projected solution are fixed, with empirical state measure $\Sigma^N_t$ as in the solution definition and \reftext{def:n-agent-fluctuation-processes-2026a}{state fluctuation process} $\mathfrak{s}^N_t=\sqrt{N}(\Sigma^N_t-S_t)$, and that the initial-condition hypotheses (I1)--(I2) of \reftext{prop:kalman-policy-cost-limit-2026a}{the cost-limit proposition} hold for these solutions: $\mathbb{E}[\mathfrak{s}^{N,\gamma}_0\mathfrak{s}^{N,\delta}_0]\to\Pi^{\gamma\delta}_0$ as $N\to\infty$ for all $\gamma,\delta\in\{1,\dots,l\}$, and $\sup\{\mathbb{E}[|\mathfrak{s}^N_0|^4]:N\ge1\}<\infty$, with the \reftext{def:expectation-variance-2026a}{expectation}. Let $J^N[h^N]$ be the \reftext{def:n-agent-cost-2026a}{$N$-agent cost} of the solution at level $N$, let $J^{MF}$ be the \reftext{def:mean-field-cost-2026a}{mean-field cost} of $(S,A)$, and set $\zeta_N=N(\mathbb{E}[\Sigma^N_0]-S_0)\in\mathbb{R}^l$ with the componentwise expectation, as in the cost-limit proposition. Then each $N(J^N[h^N]-J^{MF})+\sum_{\gamma=1}^{l}P^\gamma_0\zeta^\gamma_N$ is a well-defined real number and

\textbf{3. (Optimality of the value.)} For every \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$ with state dimension $l$, observation dimension $\tilde{l}$, and any Brownian dimension $m^\circ\ge1$, matched to the fluctuation LQG data as in hypotheses (i)--(iii) of \reftext{cor:kalman-policy-limit-is-lqg-value-2026a}{the identification corollary} --- a nonempty class of coefficient data by conclusion 3 of that corollary --- with the control dimension of the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled system} taken to be $m$, the control matrix assignment sending each $t\in[0,T]$ to $\mathcal{B}_t$, and the \reftext{def:lqg-cost-functional-2026a}{cost data} $Q_t$, $\tfrac12V_t$ (entrywise), $R_t$, $\hat{F}$, all as in the corollary, the \reftext{def:extended-lqg-cost-2026a}{linear-quadratic-Gaussian cost} $J$ attains a minimum over the \reftext{def:extended-admissible-control-2026a}{extended admissible controls} with values in $\mathbb{R}^{m}$, and

In particular, the recentred limiting cost of conclusion 2 along the approximate Kalman policies equals this minimal LQG cost.