Weighted Second-Moment Evolution of the State Fluctuation Process

Adopt the setting of the fluctuation processes of the controlled $N$-agent dynamics: 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 $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, control $\alpha_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, a mean-field trajectory pair $(S,A)$ for $\beta$ with horizon $T$, and the state fluctuation process $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$. Let $b$ be the aggregate state drift of $\beta$, let $\Theta$ be the aggregate fluctuation covariance of $\beta$, write $\mathbb{E}$ for the expectation, and set

Let $Z$ be a family of real $l\times l$ matrices $Z_t=(Z^{\gamma\delta}_t)_{\gamma,\delta\in\{1,\dots,l\}}$, $t\in[0,T]$, that is symmetric ($Z^{\gamma\delta}_t=Z^{\delta\gamma}_t$ for all $t,\gamma,\delta$) and continuously differentiable in integral form: there are continuous functions $\dot{z}^{\gamma\delta}:[0,T]\to\mathbb{R}$ — the interval regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric — with

the integral being the Riemann integral (which exists for $t>0$ by claim 3 of the integral toolkit on a compact interval applied to the restriction of $\dot{z}^{\gamma\delta}$ to $[0,t]$, continuous by claim 1 of restriction stability; it is $0$ for $t=0$). Write $\dot{z}(s)$ for the matrix $(\dot{z}^{\gamma\delta}(s))_{\gamma,\delta\in\{1,\dots,l\}}$, and for vectors $x,y\in\mathbb{R}^l$ and a real $l\times l$ matrix $M$ write $x\cdot My=\sum_{\gamma,\delta=1}^{l}M^{\gamma\delta}x^\gamma y^\delta$. Then:

(a) (Well-definedness.) For every $s\in[0,T]$ the expectations $\mathbb{E}[\mathfrak{s}_s\cdot\dot{z}(s)\mathfrak{s}_s]$, $\mathbb{E}[\mathfrak{s}_s\cdot Z_sg_s]$, and $\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]$ are finite, and as functions of $s$ they are bounded and measurable on $[0,T]$, so the Lebesgue integrals below exist.

(b) (Evolution identity.) For every $t\in[0,T]$,