Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses

Common data. Fix an 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 $R$, its transition-rate family $\beta$ with rate bound $B$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, population cost data $(L,G)$ convex in the control on $\mathcal{A}$, a twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$, a twice continuously differentiable extension of $(L,G)$, and a stationary mean-field triple $(S,A,P)$ for these data with $x_{0}=S_{0}$ in the probability simplex. Let $\Theta$ be the aggregate fluctuation covariance of $\beta$ and write $\Theta^{\star}_{s}=\Theta(S_{s},A_{s})$. Adopt the setting of the completion-of-squares theorem for these data — the matrices $E_{t}$, $\mathsf{B}_{t}$, $Q_{t}$, $V_{t}$, $R_{t}$, $\hat{F}$, the entry pairing $x\cdot My$, and hypothesis (H2) with a Riccati family $Z=(Z_{t})_{t\in[0,T]}$, which is part of the common data and does not depend on $N$. Write $\mathbb{E}$ for the expectation, $|\cdot|$ for the Euclidean norm, and $\int_{[0,T]}\cdot\,ds$ for the Lebesgue integral over a compact interval.

The family of solutions. For each natural number $N\ge1$ let there be given an $N$-agent driving system, an $\mathcal{A}$-valued observation-driven control policy $h^{(N)}$ with horizon $T$, and a solution of the controlled $N$-agent dynamics on $[0,T]$ for these data, with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$ and observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$. Write $\mathfrak{s}_{t}=\sqrt{N}(\Sigma_{t}-S_{t})$ and $\mathfrak{a}_{t}=\sqrt{N}(\alpha_{t}-A_{t})$ for its fluctuation processes, $\kappa_{0}=1+\mathbb{E}[|\mathfrak{s}_{0}|^{4}]$, and $\mathcal{J}_{N}$ for the recentred cost of the first-order expansion lemma. These objects depend on $N$; the superscript is suppressed.

Standing hypotheses. Assume that the common data and each of these solutions satisfy every standing hypothesis of the ledger lemma that does not involve the cascade parameters: hypotheses (A) and (U) of the quadratic growth lemma, hypothesis (JC) of the localized joint coercivity lemma with constant $c_{J}>0$, hypothesis (H2) above, hypothesis (LipC) of the pathwise tracking lemma, the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$ and hypothesis (TG) of the to-go comparison lemma, and the standing hypothesis on $S^{*}=S$ of the block cascade lemma. All the constants of the ledger lemma that are determined by the common data — in particular $C_{Z}$, $c_{e}$, $C_{H}$, $C_{M}$, $\bar{\Theta}$, $c_{\Theta}$, $C_{S}$, $c_{Q}$, $C_{\mathcal{D}}$, $C_{\mathcal{D}G}$, $C_{3}$, $r_{0}$, $c_{\star}$, $\rho^{*}$, $C^{\vee}_{tg}$, $C_{ns}$, $\varepsilon_{tg}$, $C_{a}$, $K_{2}$ and the moduli $\omega_{L},\omega_{b},\omega_{G}$ — are then the same for every $N$. Assume moreover:

(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 $\bigl(\mathbb{E}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]\bigr)_{N\ge1}$ has limit $\Pi^{\gamma\delta}_{0}$.

(I$'$) (Uniform initial fourth moment.) There is a real $\kappa^{\sharp}\ge1$ with $\kappa_{0}\le\kappa^{\sharp}$ for every $N\ge1$.

(CB) (Cost bound.) There is a real $\mathcal{J}^{\sharp}\ge0$ with $\mathcal{J}_{N}\le\mathcal{J}^{\sharp}$ for every $N\ge1$.

Neither (I$'$) nor (CB) constrains the fluctuation processes at positive times: (I$'$) is a hypothesis on the initial data alone, and (CB) is a hypothesis on the costs, vacuously satisfiable along any family whose costs are bounded above and carrying no information when they are not. No hypothesis bounding $\mathbb{E}[|\mathfrak{s}_{t}|^{p}]$ or $\mathbb{E}[|\mathfrak{a}_{t}|^{p}]$ uniformly in $N$ at positive times is made.

The limit value. By conclusion (a) of the completion-of-squares theorem and (H2) each entry $s\mapsto Z^{\gamma\delta}_{s}$ is continuous and bounded on $[0,T]$, and by clause (c) of the covariance deviation lemma so is each $s\mapsto\Theta^{\star\gamma\delta}_{s}$; the product is therefore continuous by continuity of sums and products and has a finite Lebesgue integral over $[0,T]$ by claim 3 of the integral toolkit. Set

a real number determined by the common data and $\Pi_{0}$.

Admissible parameter vectors. Call a tuple $\pi=(T_{0},\varepsilon_{1},\lambda_{c},\lambda_{o},\delta_{\mathrm{cl}},q_{0},\epsilon,\eta,\varrho)$ of positive reals with $T_{0}\le T$ and $\epsilon\le1$ — the clipping threshold being written $\delta_{\mathrm{cl}}$, as in the ledger lemma, because the letter $\delta$ serves throughout as a state index — admissible if, when the block cascade of the block cascade lemma and the near-field data of the ledger lemma are formed from $\pi$ for the $N$-th solution, every parameter-dependent requirement of the ledger lemma holds — namely the smallness hypothesis (SM), the requirement $\varepsilon_{1}+q_{0}\le\rho^{*}_{G}$ with $\rho^{*}_{G}$ chosen from $\epsilon$ as in the ledger lemma, and the near-field smallness condition (SN). These requirements involve only $\pi$ and constants of the common data, so admissibility does not depend on $N$.

Then the following hold.

(a) (Uniform tracked energy bound.) Let $\pi$ be admissible and let $K$ be the number of blocks it determines. Set

If in addition $\pi$ satisfies the absorption condition $2C^{\vee}_{tg}\Lambda_{\star}+2\epsilon C_{S}^{2}\le c_{\star}/4$, then there is a natural number $N_{1}$, depending on $\pi$ and on the common data but not on the solutions, such that for every $N\ge N_{1}$ the tracked energy $\mathcal{Z}$ of the $N$-th solution satisfies $\mathcal{Z}\le\mathcal{Z}^{\sharp}(\pi)$, and consequently $\mathcal{S}^{\mathrm{tr}}+\mathcal{Z}\le W^{\sharp}(\pi)$ with $\mathcal{S}^{\mathrm{tr}}$ as in the ledger lemma.

(b) (Parameter ladder.) For every real $\varepsilon'>0$ there are an admissible $\pi$ satisfying the absorption condition of (a) and a natural number $N_{0}$ such that for every $N\ge N_{0}$

the blocks $[t_{k},t_{k+1}]$, the near-field tracked events $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and the process $u$ being those of the ledger lemma formed from $\pi$ for the $N$-th solution, and the displayed sum being nonnegative.

(c) (Asymptotic lower bound.) $\displaystyle\liminf_{N\to\infty}\mathcal{J}_{N}\ \ge\ V_{0}$; that is, for every real $\varepsilon'>0$ there is a natural number $N_{0}$ with $\mathcal{J}_{N}\ge V_{0}-\varepsilon'$ for every $N\ge N_{0}$.

Conclusion (c) is the conclusion of clause (c) of the filtering lower-bound reduction lemma, with its filtering term discarded, obtained without that lemma's uniform fourth-moment hypothesis on $\mathfrak{s}$ and $\mathfrak{a}$. The filtering term retained in (b) is the object on which a lower bound by the Kalman filter covariance would act; no such bound is asserted here.