Cost Limit Along the Approximate Kalman Policy

Adopt the setting, hypotheses (H1)--(H4) and (C), and notation of the approximate Kalman filter and policy lemma — in particular the clamp indicator $\chi_t$ of its conclusion 4(a) — together with hypothesis (H5) of the filter error covariance lemma, namely that there is a real $\varrho>0$ such that every $a\in\mathbb{R}^m$ with $|a-A_t|\le\varrho$ lies in the control set $\mathcal{A}$, for every $t\in[0,T]$. Assume moreover that $\mathcal{A}$ is convex, as required by the cost expansion and completion-of-squares theorems and the filter error covariance lemma invoked in the proof; together with (C) and the boundedness of the control-side open set $V$ of the transition-rate extension, which contains $\mathcal{A}$, this makes $\mathcal{A}$ a closed, bounded, convex subset of $\mathbb{R}^m$. 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. All of this is taken for the fluctuation LQG data of the stationary mean-field triple $(S,A,P)$, whose stationary co-state $P$ has components $P^\gamma_t$ ($\gamma\in\{1,\dots,l\}$, $t\in[0,T]$): in particular the cost extension of the population cost data $(L,G)$ is part of the data, the family $Z=(Z_t)_{t\in[0,T]}$ is the Riccati family of (H2), $W_t=Z_t\mathcal{B}_t+\tfrac12V_t$, $R_t$ is symmetric positive definite with continuous inverse (conclusion 1), $\Theta^\star_t$ is the state noise covariance of the LQG data, $\Pi_0$ is the matrix of (H4), and $\Pi=(\Pi_t)_{t\in[0,T]}$ is the filter covariance of conclusion 2. For each natural number $N\ge1$, let $h^N$ be the approximate Kalman policy at level $N$, fix an $N$-agent driving system and a projected solution as in conclusion 4 of the policy lemma --- a solution of the controlled $N$-agent dynamics for $\beta$, $\tilde{\beta}$, and $h^N$ --- with empirical state measure $\Sigma^N_t$, approximate Kalman control $\alpha^N_t$, approximate Kalman filter $\hat{\mathfrak{s}}^N_t$, fluctuation processes $\mathfrak{s}^N_t=\sqrt{N}(\Sigma^N_t-S_t)$ and $\mathfrak{a}^N_t=\sqrt{N}(\alpha^N_t-A_t)$, and filter error $\varepsilon^N_t=\mathfrak{s}^N_t-\hat{\mathfrak{s}}^N_t$ with filter error covariance $\Pi^{N}_t$. Let $J^N[h^N]$ be the $N$-agent cost of this solution, let $J^{MF}=J^{MF}[(S),(A)]$ be the mean-field cost, and set $\zeta_N=N(\mathbb{E}[\Sigma^N_0]-S_0)\in\mathbb{R}^l$ with the componentwise expectation, as in the second-order cost expansion. For real matrices $A'$ and $B'$ with $l$ rows and $l$ columns write

and adopt the entry notation $x\cdot My$, $(MM')^{pq}$, $(M^{\top})^{qp}$ of the completion-of-squares theorem. Assume the two initial-condition hypotheses:

(I1) for all $\gamma,\delta\in\{1,\dots,l\}$, $\ \mathbb{E}\big[\mathfrak{s}^{N,\gamma}_0\mathfrak{s}^{N,\delta}_0\big]\to\Pi^{\gamma\delta}_0$ as $N\to\infty$;

(I2) $\ \sup\big\{\mathbb{E}\big[|\mathfrak{s}^N_0|^4\big]:N\ge1\big\}<\infty$.

Then the map $t\mapsto Z_t\cdot\Theta^\star_t+(W_tR_t^{-1}W_t^{\top})\cdot\Pi_t$ is continuous on $[0,T]$, 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

the integral being the Lebesgue integral over the compact interval $[0,T]$ of the continuous integrand (which agrees with its Riemann integral by claim 3 of that toolkit).