Mean-Square Error Covariance of the Approximate Kalman Filter

Adopt the setting, hypotheses \textbf{(H1)}--\textbf{(H4)}, and notation of the \reftext{lem:approximate-kalman-policy-2026a}{approximate Kalman filter and policy lemma}: the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data} of the \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple} $(S,A,P)$ with matrices $\mathcal{E}_t$, $\mathcal{B}_t$, $\tilde{\mathcal{E}}_t$, $\Theta^\star_t$, $\tilde{\Theta}^\star_t$, the coefficient matrices $Q_t$, $V_t$, $R_t$, $W_t=Z_t\mathcal{B}_t+\tfrac12V_t$, the feedback gain $\mathcal{G}_t=R_t^{-1}W_t^{\top}$, the filter covariance $\Pi=(\Pi_t)_{t\in[0,T]}$ with initial value $\Pi_0$ from (H4), the Kalman gain $\tilde{\mathcal{K}}_t=\Pi_t\tilde{\mathcal{E}}_t^{\top}(\tilde{\Theta}^\star_t)^{-1}$, and $\tilde{D}_t=\tilde{\mathcal{E}}_t^{\top}(\tilde{\Theta}^\star_t)^{-1}\tilde{\mathcal{E}}_t$. Fix a natural number $N\ge1$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} $(\Omega,\mathcal{F},\mathbb{P})$, and a projected \reftext{def:n-agent-controlled-dynamics-2026a}{solution} as in conclusion 4 of the policy lemma: state processes $\sigma^i$, observation processes $\Upsilon^\upsilon$, the approximate Kalman control $\alpha$, the approximate Kalman filter $\hat{\mathfrak{s}}^N$, the regular event $\Omega_0$, the system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ (the system filtration of the \reftext{def:n-agent-controlled-dynamics-2026a}{solution definition}), the empirical state measure $\Sigma_t$, and the observation total $\tilde{c}_t$ with jump times $\tau_j$ and channels $\upsilon_j$; this collection is a solution of the controlled $N$-agent dynamics for $\beta$, $\tilde{\beta}$, and the approximate Kalman policy $h^N$, by conclusion 4(a). Since $(S,A)$ is a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} (part of the \reftext{def:stationary-mean-field-triple-2026a}{stationary triple}), the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes} $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$ are defined; by conclusion 4(a) of the policy lemma, $\mathfrak{a}_t=-\mathcal{G}_t\,\hat{\mathfrak{s}}^N_t$ at every point of $\Omega_0\times[0,T]$. Let $e_s$ be the \reftext{lem:fluctuation-linearization-residual-2026a}{state linearization residual} of this solution about $(S,A)$ (whose linearization matrices $E_s$ and $\mathsf{B}_s$ coincide with $\mathcal{E}_s$ and $\mathcal{B}_s$, by the identical defining formulas of clauses 1 and 2 of the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data}), let $\tilde{e}_s$ be the \reftext{lem:observation-linearization-residual-2026a}{observation linearization residual} (whose observation linearization matrix coincides with $\tilde{\mathcal{E}}_s$, as noted there), let $M_t$ be the state martingale vector of the \reftext{thm:n-agent-martingale-decomposition-2026a}{martingale decomposition} (not to be confused with the closed-loop matrix $\mathcal{M}_t=\mathcal{E}_t-\mathcal{B}_t\mathcal{G}_t-\tilde{\mathcal{K}}_t\tilde{\mathcal{E}}_t$ used in the proof below, written $M_t$ in the \reftext{lem:approximate-kalman-policy-2026a}{policy lemma}), and let

be the \reftext{lem:n-agent-weighted-observation-sums-2026a}{weighted compensated observation sum} with weight $F=\tilde{\mathcal{K}}$ (admissible there: all entries of $t\mapsto\tilde{\mathcal{K}}_t$ are continuous by conclusion 2 of the policy lemma; fix a real $C_{\tilde{K}}\ge0$ bounding them in absolute value, by \reftext{lem:continuous-compact-interval-bounded-2026a}{boundedness of continuous functions}). Write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, $|\cdot|$ for the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), $\mathbf{1}_{\Omega_0}$ for the indicator of $\Omega_0$, and adopt the matrix entry notation of the \reftext{thm:fluctuation-control-coercivity-2026b}{completion-of-squares theorem}: $x\cdot My=\sum_{p,q}M^{pq}x^py^q$, $(MM')^{pq}=\sum_{r}M^{pr}M'^{rq}$, $(M^{\top})^{qp}=M^{pq}$. Let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$ and $\tilde{b}$ the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$. Define the \textbf{filter error}

and set

which is finite because $|\mathfrak{s}_0|\le2\sqrt{N}$ everywhere (any two points of the \reftext{def:probability-simplex-2026a}{probability simplex} being at Euclidean distance at most $2$). Then:

\textbf{(a) (Finiteness and well-definedness.)} The control energies

are finite (they are well defined in $[0,\infty]$ by clause (a) of the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound} and of the \reftext{lem:fluctuation-fourth-moment-bound-2026a}{a priori fourth-moment bound}); in particular the \reftext{lem:fluctuation-linearization-residual-2026a}{state residual lemma} applies to this solution. For every $t$, each $\varepsilon^\gamma_t$ is a \reftext{def:probability-space-random-variable-2026a}{random variable}, measurable with respect to $\mathcal{F}^{\mathrm{sys}}_t$, with $\mathbb{E}[|\varepsilon_t|^p]<\infty$ for every natural number $p\ge1$; each map $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\varepsilon^\gamma_t(\omega)$ is \reftext{def:measurable-function-2026a}{measurable} for the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} of the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and $\mathcal{F}$; and the \textbf{filter error covariance}

is finite, symmetric in $(\gamma,\delta)$, measurable and bounded as a function of $t$, with $\Pi^{N,\gamma\delta}_0=\mathbb{E}[\mathfrak{s}^\gamma_0\mathfrak{s}^\delta_0]$ (the filter starting at $\hat{\mathfrak{s}}^N_0=0$).

\textbf{(b) (Error equation.)} \reftext{def:almost-surely-2026a}{Almost surely}, for every $t\in[0,T]$, componentwise,

all integrals existing componentwise at the relevant outcomes (the integral being $0$ for $t=0$).

\textbf{(c) (Covariance evolution.)} Define, for $r\in[0,T]$ and $\gamma,\delta\in\{1,\dots,l\}$,

All expectations below are finite, all integrands are bounded and measurable in $r$, and for every $t\in[0,T]$:

In particular each $t\mapsto\Pi^{N,\gamma\delta}_t$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous}.

\textbf{(d) (Uniform moment bounds.)} There is a real $C_1\ge0$, determined by $l$, $\tilde{l}$, $m$, $T$, the rate bounds $B$ and $\tilde{B}$, the derivative bounds $K$ and $\tilde{K}$ (of the \reftext{def:c2-transition-rate-extension-2026a}{transition-rate} and \reftext{def:c2-observation-rate-extension-2026a}{observation-rate} extensions respectively), and the entry bounds of the continuous matrix families $\mathcal{E}$, $\mathcal{B}$, $\tilde{\mathcal{E}}$, $\mathcal{G}$, $\tilde{\mathcal{K}}$ on $[0,T]$ --- in particular the same for every $N$, every driving system, and every projected solution --- such that

and consequently, using $\mathbb{E}[|\cdot|^2]\le1+\mathbb{E}[|\cdot|^4]$ and $\kappa_0\ge1$, also $\sup_t(\mathbb{E}[|\mathfrak{s}_t|^2]+\mathbb{E}[|\hat{\mathfrak{s}}^N_t|^2]+\mathbb{E}[|\varepsilon_t|^2]+\mathbb{E}[|\mathfrak{a}_t|^2])\le(4+C_1)\,\kappa_0$ and $\mathcal{A}+\mathcal{A}_4\le2T(4+C_1)\,\kappa_0$.

\textbf{(e) (Covariance comparison.)} There is a real $C\ge0$, determined by the same data as $C_1$ together with the entry bounds of $\Pi$ and of $(\tilde{\Theta}^\star)^{-1}$ on $[0,T]$ --- again not depending on $N$, on the driving system, or on the solution --- such that for every $t\in[0,T]$ and all $\gamma,\delta\in\{1,\dots,l\}$: