Mean-Square Error Covariance of the Approximate Kalman Filter

Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma: the fluctuation LQG data of the 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 $N$-agent driving system $(\Omega,\mathcal{F},\mathbb{P})$, and a projected 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 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 mean-field trajectory pair (part of the stationary triple), the 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, $\mathfrak{a}_t=-\chi_t\,\mathcal{G}_t\,\hat{\mathfrak{s}}^N_t$ at every point of $\Omega_0\times[0,T]$, where $\chi_t$ is the clamp indicator of that conclusion, equal to $1$ at those $(t,\omega)$ at which $A_t-N^{-1/2}\mathcal{G}_t\hat{\mathfrak{s}}^N_t$ lies in the control set $\mathcal{A}$ and to $0$ otherwise. Define the clamp remainder

which vanishes at every point at which the clamp is inactive; it is the exact defect in the cancellation $\mathcal{B}_t\mathfrak{a}_t+\mathcal{B}_t\mathcal{G}_t\hat{\mathfrak{s}}^N_t=w_t$ of the control terms of the state and filter equations. Let $e_s$ be the 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 fluctuation LQG data), let $\tilde{e}_s$ be the 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 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 policy lemma), and let

be the 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, their absolute values attaining a maximum by the extreme value theorem). Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Write $\mathbb{E}$ for the expectation, $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), $\mathbf{1}_{\Omega_0}$ for the indicator of $\Omega_0$, and adopt the matrix entry notation of the 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 aggregate fluctuation covariance of $\beta$ and $\tilde{b}$ the aggregate observation drift of $\tilde{\beta}$. Define the filter error

and set

which is finite because $|\mathfrak{s}_0|\le2\sqrt{N}$ everywhere (any two points of the probability simplex being at Euclidean distance at most $2$). Assume in addition:

(H5) (Interior mean-field control.) There is a real $\varrho>0$ such that for every $t\in[0,T]$ every $a\in\mathbb{R}^m$ with $|a-A_t|\le\varrho$ lies in the control set $\mathcal{A}$. (The letter $\varrho$ is used for this constant because $\delta$ is reserved throughout for a matrix index.)

(H6) (Convex control set.) The control set $\mathcal{A}$ is convex.

Then:

(a) (Finiteness and well-definedness.) The control energies

are finite (they are well defined in $[0,\infty]$ by clause (a) of the a priori second-moment bound and of the a priori fourth-moment bound); in particular the state residual lemma applies to this solution. For every $t$, each $\varepsilon^\gamma_t$ is a 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 measurable for the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$; and the 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 $\hat{\mathfrak{s}}^N_0=0$).

(b) (Error equation.) Almost surely, for every $t\in[0,T]$, componentwise,

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

(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 continuous on $[0,T]$.

(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 transition-rate and 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 $\mathcal{A}_2+\mathcal{A}_4\le2T(4+C_1)\,\kappa_0$.

(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 $[0,T]$ and the constant $\varrho$ of (H5) --- again not depending on $N$, on the driving system, or on the solution --- such that for every $t\in[0,T]$ and all $\gamma,\delta\in\{1,\dots,l\}$: