Filtering Lower-Bound Reduction of the Recentred N-Agent Cost

Fix the following common data: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with $\tilde{l}$ channels, a horizon $T>0$, a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} $(U,\bar{\beta})$ of $\beta$ with derivative bound $K$, a \reftext{def:c2-population-cost-extension-2026b}{twice continuously differentiable extension} of \reftext{def:population-cost-data-2026a}{population cost data} $(L,G)$ with second-derivative bound $K_c$, a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$ for $\beta$ with horizon $T$, and a stationary co-state $P$ for these data, so that $(S,A,P)$ is a \reftext{def:stationary-mean-field-triple-2026b}{stationary mean-field triple}. Let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$ and write $\Theta^\star_s=\Theta(S_s,A_s)$.

For each \reftext{def:natural-numbers-2026a}{natural number} $N\ge1$, let there be given an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system}, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h^{(N)}$ with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, and a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} of the controlled $N$-agent dynamics on $[0,T]$ for these data, with regular event, empirical state measure $\Sigma^{(N)}_t$, control $\alpha^{(N)}_t$, and observation filtration $(\mathcal{G}^{(N)}_t)_{t\in[0,T]}$ as in the solution definition. Adopt, for the $N$-th solution, the setting and notation of \reftext{thm:fluctuation-control-coercivity-2026b}{the completion-of-squares theorem}: the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes} $\mathfrak{s}^{(N)}_t$ and $\mathfrak{a}^{(N)}_t$, the \reftext{def:n-agent-cost-2026a}{$N$-agent cost} $J^N[h^{(N)}]$, the \reftext{def:mean-field-cost-2026a}{mean-field cost} $J^{MF}$, the vector $\zeta_N$, remainder $R_N$, \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} $d$ and norm $|\cdot|$ of \reftext{thm:n-agent-cost-expansion-2026b}{the second-order expansion}, the \reftext{def:fluctuation-lqg-cost-2026b}{fluctuation linear-quadratic cost} $LQG[(\mathfrak{s}^{(N)}),(\mathfrak{a}^{(N)})]$, the coefficient matrices $E_t$, $\mathsf{B}_t$, $Q_t$, $V_t$, $R_t$, $\hat{F}$, the entry pairing $x\cdot My$, the quantities $g_s$ and $e_s=g_s-E_s\mathfrak{s}^{(N)}_s-\mathsf{B}_s\mathfrak{a}^{(N)}_s$, and the constants $C_P$, $C_Z$, $C_K$, $c_e$ of those theorems, which depend only on the common data (the letter $\Lambda$ of the adopted setting retains its meaning there as a scalar constant and is not used below). Assume hypotheses \textbf{(H1)}--\textbf{(H2)} of the completion-of-squares theorem, with the Riccati family $Z=(Z_t)_{t\in[0,T]}$ and $W_t=Z_t\mathsf{B}_t+\tfrac12V_t$ fixed throughout, and set, for the $N$-th solution, $u^{(N)}_t=\mathfrak{a}^{(N)}_t+R_t^{-1}W_t^T\mathfrak{s}^{(N)}_t$ as there, and

Assume moreover, with $\mathbb{E}$ the \reftext{def:expectation-variance-2026a}{expectation}:

\textbf{(M) (Uniform fourth moments.)} There is a real $M\ge0$ such that for every $N\ge1$ and every $t\in[0,T]$, $\mathbb{E}\big[|\mathfrak{s}^{(N)}_t|^4\big]\le M$ and $\mathbb{E}\big[|\mathfrak{a}^{(N)}_t|^4\big]\le M$, the expectations of these nonnegative random variables being taken in $[0,\infty]$.

\textbf{(I) (Initial covariance convergence.)} There is a symmetric real matrix $\Pi_0$ with $l$ rows and $l$ columns such that for all $\gamma,\delta\in\{1,\dots,l\}$ the real sequence $\big(\mathbb{E}[\mathfrak{s}^{(N),\gamma}_0\mathfrak{s}^{(N),\delta}_0]\big)_{N\ge1}$ has \reftext{def:limit-sequence-real-c54-2026a}{limit} $\Pi^{\gamma\delta}_0$.

Then:

\textbf{(a) (Applicability and vanishing error.)} For every $N$ and $t$, $\mathbb{E}[|\mathfrak{a}^{(N)}_t|^2]\le\tfrac12(1+M)$ and $\mathbb{E}[|\mathfrak{s}^{(N)}_t|^2]\le\tfrac12(1+M)$; in particular $\mathcal{A}_N=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}^{(N)}_t|^2]\,dt\le\tfrac{T}{2}(1+M)<\infty$, so \reftext{thm:n-agent-cost-expansion-2026b}{the second-order expansion} and \reftext{thm:fluctuation-control-coercivity-2026b}{the completion-of-squares theorem} apply to the $N$-th solution. Moreover the real number $r_N$ defined by the identity

in which every term is a well-defined real number --- the last integral being the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,T]$ of a bounded measurable function by clause (c) of \reftext{lem:fluctuation-covariance-deviation-2026a}{the covariance deviation lemma} --- satisfies: the real sequence $(r_N)_{N\ge1}$ has \reftext{def:limit-sequence-real-c54-2026a}{limit} $0$.

\textbf{(b) (Filtering bound.)} Set $\Xi_t=W_tR_t^{-1}W_t^T$, a symmetric \reftext{def:positive-semidefinite-matrix-2026a}{positive semidefinite} real matrix with $l$ rows and $l$ columns for every $t\in[0,T]$. For every $N\ge1$ and every $t\in[0,T]$: the components of $\mathfrak{s}^{(N)}_t$ and of $\mathfrak{a}^{(N)}_t$ are \reftext{def:square-integrable-mean-square-2026a}{square-integrable}; each component of $\mathfrak{a}^{(N)}_t$ is \reftext{def:almost-surely-2026a}{almost surely} equal to a $\mathcal{G}^{(N)}_t$-measurable square-integrable random variable, by \reftext{lem:n-agent-control-observation-adapted-2026a}{the observation-adaptedness lemma}; and for every choice of \reftext{def:conditional-expectation-l2-2026a}{conditional expectations} $\mathbb{E}[\mathfrak{s}^{(N),\gamma}_t|\mathcal{G}^{(N)}_t]$ ($\gamma\in\{1,\dots,l\}$), which exist by \reftext{thm:conditional-expectation-l2-2026a}{the existence and uniqueness theorem}, the filtering error $\varepsilon^{(N)}_t$ with components $\varepsilon^{(N),\gamma}_t=\mathfrak{s}^{(N),\gamma}_t-\mathbb{E}[\mathfrak{s}^{(N),\gamma}_t|\mathcal{G}^{(N)}_t]$ satisfies

the middle quantity being independent of the choice of conditional expectations.

\textbf{(c) (Lower bound.)} For every real $\varepsilon'>0$ there is a natural number $N_0$ such that for every $N\ge N_0$:

Conclusion (b) bounds the integrand of the last integral pointwise in $t$; conclusion (c) is deliberately stated in terms of that integral, no measurability in $t$ of the filtering-error term being asserted.