Second-Order Expansion of the N-Agent Cost about a Stationary Mean-Field Trajectory

Adopt the setting of the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes of the controlled $N$-agent dynamics}: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$, a horizon $T>0$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system}, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h$, a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$, a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$ for $\beta$ with horizon $T$, and the fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Let $(L,G)$ be \reftext{def:population-cost-data-2026a}{population cost data} on $l$ states with control dimension $m$, with \reftext{def:n-agent-cost-2026a}{$N$-agent cost} $J^N[h]$ and \reftext{def:mean-field-cost-2026a}{mean-field cost} $J^{MF}[(S),(A)]$. Let $(U,\bar{\beta})$ be a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} of $\beta$ with derivative bound $K$ and extended aggregate state drift $\bar{b}$, let $(V,\bar{L},\bar{G})$ be a \reftext{def:c2-population-cost-extension-2026a}{twice continuously differentiable extension} of $(L,G)$ with second-derivative bound $K_c$, and let $P$ be a \reftext{def:stationary-mean-field-triple-2026a}{stationary co-state} for these data, so that $(S,A,P)$ is a stationary mean-field triple. Adopt the partial-derivative notation $\partial_j\partial_i$ of the extension definitions, write $d$ for the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}, $|\cdot|$ for the Euclidean norm, $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, and $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}, and let $C_P$ be a real number with $\sum_{\delta=1}^{l}|P^\delta_t|\le C_P$ for all $t\in[0,T]$, which exists because each component of $P$ is continuous and hence \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}.

\textbf{Hypothesis.} Assume $\mathcal{A}=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}_t|^2]\,dt<\infty$, this integral being well defined by part (a) of the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound}.

For $u\ge0$ define

\textbf{(a) (Moduli.)} $\omega_L$, $\omega_b$, and $\omega_G$ are nondecreasing functions from $[0,\infty)$ to $[0,\infty)$ with $\omega_L\le2K_c$, $\omega_b\le6\,l\,K$, and $\omega_G\le2K_c$ everywhere, and for every $\varepsilon>0$ there is $\delta>0$ such that $\omega_L(u)\le\varepsilon$, $\omega_b(u)\le\varepsilon$, and $\omega_G(u)\le\varepsilon$ for all $u\in[0,\delta]$.

\textbf{(b) (Finiteness and eligibility.)} $J^N[h]$ is finite; the pair $((\mathfrak{s}),(\mathfrak{a}))$ satisfies requirements (i)-(iii) of the \reftext{def:fluctuation-lqg-cost-2026a}{fluctuation linear-quadratic cost} (requirement (i) holding with $\Omega_1=\Omega_0$ by the \reftext{lem:n-agent-joint-measurability-2026a}{joint measurability of the state and control}), so that $LQG[(\mathfrak{s}),(\mathfrak{a})]$ is a well-defined real number; and all expectations and integrals appearing in (c) are well defined and finite.

\textbf{(c) (Expansion with quantitative remainder.)} Set $\zeta_N=N\,(\mathbb{E}[\Sigma_0]-S_0)\in\mathbb{R}^l$, with the componentwise expectation, and set $\rho_t=d\big((\Sigma_t,\alpha_t),(S_t,A_t)\big)$ for $t\in[0,T]$, so that $\rho_t=N^{-1/2}\big(|\mathfrak{s}_t|^2+|\mathfrak{a}_t|^2\big)^{1/2}$ and $d(\Sigma_T,S_T)=N^{-1/2}|\mathfrak{s}_T|$. Then the real number $R_N$ defined by the identity

satisfies