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

Fix the following common data: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, and rate bound $B$, an observation-rate family $\tilde{\beta}$ with $\tilde{l}$ channels, a horizon $T>0$, a twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$ with derivative bound $K$, a twice continuously differentiable extension $(U_c,\bar{L},\bar{G})$ of population cost data $(L,G)$ with second-derivative bound $K_c$ — its open set, written $W$ in the second-order expansion theorem below, is written $U_c$ here as in the completion-of-squares theorem, the letter $W$ being reserved for the matrices $W_t$ —, a 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 stationary mean-field triple. Assume the control set $\mathcal{A}$ is convex. Let $\Theta$ be the aggregate fluctuation covariance of $\beta$ and write $\Theta^\star_s=\Theta(S_s,A_s)$.

For each natural number $N\ge1$, let there be given an $N$-agent driving system, an observation-driven control policy $h^{(N)}$ with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, which is $\mathcal{A}$-valued, and a 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 the completion-of-squares theorem: the fluctuation processes $\mathfrak{s}^{(N)}_t$ and $\mathfrak{a}^{(N)}_t$, the $N$-agent cost $J^N[h^{(N)}]$, the mean-field cost $J^{MF}$, the vector $\zeta_N$, remainder $R_N$, Euclidean distance $d$ and norm $|\cdot|$ of the second-order expansion, the 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 time-indexed matrices $V_t$ and $W_t$ being unrelated to the control-side open set $V$ of $(U,V,\bar{\beta})$, and the sans-serif $\mathsf{B}_t$ distinct from the rate bound $B$ —, 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, none of which depends on $N$ — $C_P$ and $c_e$ depend only on the common data, and $C_Z$ and $C_K$ also on the Riccati family $Z$ fixed in hypothesis (H2) below, itself independent of $N$ — (the letters $\Lambda$ and $\hat{\Lambda}$ of the adopted setting retain their meanings there as scalar constants and are not used below). 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. Assume hypotheses (H1)--(H2) of the completion-of-squares theorem — its integrability hypothesis $\mathcal{A}_2<\infty$, the subscripted $\mathcal{A}_2$ being distinct from the control set $\mathcal{A}$, is verified, for each $N$, in conclusion (a) below —, 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 expectation:

(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 $[0,\infty]$.

(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 limit $\Pi^{\gamma\delta}_0$.

Then:

(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 the energy $\mathcal{A}^{(N)}_2=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}^{(N)}_t|^2]\,dt$ of the adopted setting (the quantity $\mathcal{A}_2$ there) satisfies $\mathcal{A}^{(N)}_2\le\tfrac{T}{2}(1+M)<\infty$, so every hypothesis of the second-order expansion and of the completion-of-squares theorem holds for the $N$-th solution, and both apply to it. 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 Lebesgue integral over the compact interval $[0,T]$ of a bounded measurable function: each $s\mapsto\Theta^{\star\gamma\delta}_s$ is continuous and bounded by clause (c) of the covariance deviation lemma, each entry $s\mapsto Z^{\gamma\delta}_s$ is continuous and bounded by conclusion (a) of the completion-of-squares theorem and hypothesis (H2), the integrand is then continuous by continuity of sums and products, and a continuous function on $[0,T]$ is measurable with well-defined finite Lebesgue integral by clause 3 of the toolkit --- satisfies: the real sequence $(r_N)_{N\ge1}$ has limit $0$.

(b) (Filtering bound.) Set $\Xi_t=W_tR_t^{-1}W_t^T$, a symmetric 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 square-integrable; each component of $\mathfrak{a}^{(N)}_t$ is almost surely equal to a $\mathcal{G}^{(N)}_t$-measurable square-integrable random variable, by the observation-adaptedness lemma; and for every choice of conditional expectations $\mathbb{E}[\mathfrak{s}^{(N),\gamma}_t|\mathcal{G}^{(N)}_t]$ ($\gamma\in\{1,\dots,l\}$), which exist by 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.

(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.