Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound

Adopt the setting, notation, parameters and standing hypotheses of the tracked energy bound lemma, formed for one and the same data as there. In particular: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and control bound $R$, and its transition-rate family $\beta$ with rate bound $B$; the horizon $T>0$; the population cost data $(L,G)$ with its twice continuously differentiable extension, and the twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$ (whose derivative bound is not used below, so that the letter $K$ is free here for the number of blocks); the solution of the controlled $N$-agent dynamics with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$; the stationary mean-field triple $(S,A,P)$ with the bound $C_{P}$; the fluctuation processes $\mathfrak{s}$, $\mathfrak{a}$ and $\mathfrak{z}=(\mathfrak{s},\mathfrak{a})$; the realized mean-field flow $\Phi$, the deviation $Y$, the energy $\mathcal{E}$, the real $q_{0}>0$, and the clipping threshold of the extended good-set stopping-time lemma, written $\delta$ there and $\delta_{\mathrm{cl}}$ here because the letter $\delta$ is reserved below for a state index; the noise majorant $Q$ with $\mathbb{E}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$ and the noise event $\mathcal{N}=\{Q>q_{0}\}$; the block data $T_{0}$, $K$, $t_{k}$, $h_{k}$, the levels $L_{k}$ and the reals $\varepsilon_{1}$, $\lambda_{c}$, $\lambda_{o}$, the constants $C_{a}$, $\Lambda_{\star}$ and $C_{S}$, the anchored clocks $\sigma^{(k)}$, the good sets $G_{k}$, the leave events $D_{k}$, the block energy increments $\Delta_{k}\mathcal{E}$ and the tracked energy $\mathcal{Z}=N\sum_{k=0}^{K-1}\mathbb{E}[\mathbf{1}_{G_{k}}\Delta_{k}\mathcal{E}]$; the quantities $\mathcal{D}_{t}$, $\mathcal{D}_{G}$ and $\mathcal{J}_{N}$ with the bounds $C_{\mathcal{D}}$, $C_{\mathcal{D}G}$ and the constant $C_{3}$ of the first-order expansion lemma for the recentred $N$-agent cost; the constants $c_{J}$, $\rho^{*}$, $r_{0}$, $c_{\star}$, $C^{\vee}_{tg}$, $C_{ns}$, $\varepsilon_{tg}$, the real $\epsilon>0$ and the radius $\rho^{*}_{G}$; and the moduli of continuity $\omega_{L}$, $\omega_{b}$, $\omega_{G}$ and the radius $\rho_{t}=d((\Sigma_{t},\alpha_{t}),(S_{t},A_{t}))$ of the second-order expansion theorem, $d$ being the Euclidean distance and $|\cdot|$ the Euclidean norm. Adopt likewise the standing hypotheses (A), (U), (JC) with constant $c_{J}>0$, (LipC), the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$, (TG), the standing hypothesis on $S^{*}=S$ of the block cascade lemma, the smallness hypothesis (SM), and the additional requirement $\varepsilon_{1}+q_{0}\le\rho^{*}_{G}$ imposed in claim 4 of the tracked energy bound lemma.

Adopt in addition the setting and notation of the stopped completion-of-squares lemma and of the cascade filtering lemma, formed for the same data, and with them the setting of the completion-of-squares theorem: the fluctuation Hessian coefficients $H_{ij}(t)$ ($i,j\in\{1,\dots,l+m\}$) and terminal coefficients $F_{\gamma\delta}$; the matrices $E_{t}$, $\mathsf{B}_{t}$, $Q_{t}$, $V_{t}$, $R_{t}$ and $\hat{F}$ and the entry pairing $x\cdot My=\sum_{p,q}M^{pq}x^{p}y^{q}$ defined there; hypotheses (H1) and (H2) with the Riccati family $Z=(Z_{t})_{t\in[0,T]}$, $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac{1}{2}V_{t}$ and $Z_{T}=\hat{F}$; the processes $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{T}\mathfrak{s}_{t}$ and $e_{s}=g_{s}-E_{s}\mathfrak{s}_{s}-\mathsf{B}_{s}\mathfrak{a}_{s}$; the constants $C_{Z}$ and $c_{e}$; and, from the cascade filtering lemma, the matrix family $\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{T}$ and the tracked events of its claim 1. Adopt also the aggregate fluctuation covariance $\Theta$ of $\beta$, write $\Theta^{\star}_{s}=\Theta(S_{s},A_{s})$, and adopt the constant $c_{\Theta}$ of the covariance deviation lemma. Hypothesis (H1) holds, with $r=c_{J}$, by claim 5 of the localized joint coercivity lemma under (JC); hypothesis (H2) is assumed here.

Two notational collisions are resolved as follows. The matrices $Q_{t}$ of the completion-of-squares theorem always carry their time subscript and are distinct from the noise majorant $Q$, which never carries one; likewise $V_{t}$ is distinct from the control-side open set $V$ of the extension $(U,V,\bar{\beta})$, and $R_{t}$ from the control bound $R$. The quantity written $\Xi_{K}$ in claim 6 of the block cascade lemma is here written $\Upsilon_{\mathrm{esc}}$, the letter $\Xi$ being reserved for the matrix family $\Xi_{t}$ above.

Near-field parameters. Let $\eta>0$ and $\varrho>0$ be real numbers, and let $\rho_{\eta}>0$ be a real number such that

which exists by part (a) of the second-order expansion theorem, exactly as the radius of claim 2 of the localized joint coercivity lemma is obtained there. Assume the near-field smallness condition

The radius $\rho^{*}_{G}$ is fixed here more precisely than in the tracked energy bound lemma: let $\rho^{*}_{G}>0$ be a real number with $\tfrac{1}{2}\,l\,\omega_{G}(v)\le\epsilon$ for every $v\in[0,\rho^{*}_{G}]$, which exists by part (a) of the second-order expansion theorem. By the terminal half of claim 1 of the localized joint coercivity lemma such a number has the property demanded of the radius in claim 4 of that lemma, so this is an admissible choice of the $\rho^{*}_{G}$ appearing in claim 4 of the tracked energy bound lemma.

Constants. Each of the families $t\mapsto H_{ij}(t)$, $t\mapsto W^{\gamma j}_{t}$, $t\mapsto V^{\gamma j}_{t}$, $t\mapsto Q^{\gamma\delta}_{t}$, $t\mapsto(W_{t}R_{t}^{-1}W_{t}^{T})^{\gamma\delta}$ and $s\mapsto\Theta^{\star\gamma\delta}_{s}$ is continuous on $[0,T]$ — the first by the extension definitions together with the continuity of $t\mapsto(S_{t},A_{t})$ and of $t\mapsto P_{t}$, the next four by conclusion (a) of the completion-of-squares theorem and (H2), and the last by clause (c) of the covariance deviation lemma — and hence bounded there by the extreme value theorem. Fix reals $C_{H}\ge0$, $C_{M}\ge0$ and $\bar{\Theta}\ge0$ bounding, in absolute value, the first family, the next four families, and the last family respectively, for all indices and all points of $[0,T]$, and set

Derived quantities. Write $\mathbf{1}_{D}$ for the indicator of a set $D$, $\mathbb{E}$ for the expectation, $P$ for the probability of the driving system (the stationary co-state, also written $P$, always carries a time subscript), $\int_{[a,b]}\cdot\,ds$ for the Lebesgue integral over a compact interval, and, for a nonnegative real $x$, $x^{1/2}$ for the nonnegative square root and $x^{3/4}=(x^{1/4})^{3}$ as in the restricted moments lemma. For $k\in\{0,\dots,K-1\}$ and $s\in[0,T]$ set

so that $\mathcal{T}_{k}(s)$ is the union of the disjoint sets $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$; and set

the sampled functions being those of the sampled-process definition, taken componentwise, and

Finally set

Then the following hold.

1. (Tracked moments and escape sums.) All the quantities below are finite, and:

(a) $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{a}_{s}|^{2}\bigr]ds=\mathcal{Z}$.

(b) $\mathcal{S}^{\mathrm{tr}}\le 2\,C_{S}^{2}\,T\,\mathcal{Z}+2\,T\,c_{Q}^{1/2}\kappa_{0}^{1/2}$.

(c) $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}\bigr]ds\le\frac{\mathcal{Z}}{N\varrho^{2}}$ and $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}|\mathfrak{a}_{s}|\bigr]ds\le\frac{\mathcal{Z}}{\sqrt{N}\,\varrho}$.

(d) $\mathcal{P}\le\Lambda_{\star}\mathcal{Z}\,\Upsilon_{\mathrm{lev}}\,N^{-1}$, and $\displaystyle\sum_{k=0}^{K-1}\sqrt{N}\,P(D_{k})^{3/4}\le(\Lambda_{\star}\mathcal{Z})^{3/4}N^{-1/4}\,\Upsilon_{\mathrm{esc}}$, and $\displaystyle\sum_{k=0}^{K-1}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[1-\mathbf{1}_{\mathcal{T}_{k}(s)}\bigr]ds\le T\,\mathcal{P}$.

(e) $\displaystyle\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{D_{k}}|\mathfrak{s}_{\sigma^{(k)}}|^{2}\bigr]\le 2\Lambda_{\star}\mathcal{Z}+2\,c_{Q}^{1/2}\kappa_{0}^{1/2}\,\mathcal{P}^{1/2}$.

2. (The near-field tracked events are observable.) For every $k\in\{0,\dots,K-1\}$ and every $s\in[t_{k},T]$ the set $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ is an event belonging to $\mathcal{G}_{s}$; consequently, by claim 3 of the cascade filtering lemma, for any choice of conditional expectations defining the filtering error $\varepsilon_{s}$ there,

3. (Master ledger identity.) With $\int_{[\varsigma,T]}X_{t}\,dt$ as in the post-exit comparison lemma,

all terms being finite.

4. (Leaver bill.)

5. (Terminal bill.)

6. (Block bound with the far field cancelled.) For every $k\in\{0,\dots,K-1\}$,

where

7. (Assembled lower bound.)

and in particular, the last sum being nonnegative by claim 2,

No hypothesis on the moments of $\mathfrak{s}$ or $\mathfrak{a}$, uniform in $N$ or otherwise, is used.