Stopped Weighted Second-Moment Evolution of the State Fluctuation Process

Adopt the setting of the weighted second-moment evolution lemma for the state fluctuation process: 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 $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ — together with the aggregate state drift $b$ and the aggregate fluctuation covariance $\Theta$ of $\beta$, the drift difference $g_s=\sqrt{N}\,(b(\Sigma_s,\alpha_s)-b(S_s,A_s))\in\mathbb{R}^l$, and the weight family $Z=(Z_t)_{t\in[0,T]}$ of symmetric real $l\times l$ matrices, continuously differentiable in integral form with derivative matrices $\dot{z}(s)$, exactly as specified there, with the pairing notation $x\cdot My=\sum_{\gamma,\delta=1}^{l}M^{\gamma\delta}x^\gamma y^\delta$ and with $\mathbb{E}$ the expectation.

Additionally, let $\tau$ be a stopping time of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$. Write $\mathbf{1}_A$ for the function equal to $1$ on $A$ and $0$ off $A$; for $s\in[0,T]$, $\mathbf{1}_{\{s<\tau\}}$ denotes the value at time $s$ of the pre-stopping-time indicator of $\tau$, the function equal to $1$ on $\{\omega\in\Omega:s<\tau(\omega)\}$ and $0$ off it. For $t\in[0,T]$, $\min(t,\tau)$ denotes the function $\omega\mapsto\min(t,\tau(\omega))$, and the sampled function at $\min(t,\tau)$ is applied componentwise: $\mathfrak{s}_{\min(t,\tau)}$ is the $\mathbb{R}^l$-valued function on $\Omega$ with components $\mathfrak{s}^\gamma_{\min(t,\tau)}(\omega)=\mathfrak{s}^\gamma_{\min(t,\tau(\omega))}(\omega)$, and $Z_{\min(t,\tau)}$ is the matrix-valued function on $\Omega$ with entries $Z^{\gamma\delta}_{\min(t,\tau)}(\omega)=Z^{\gamma\delta}_{\min(t,\tau(\omega))}$. Then:

(a) (Well-definedness.) For every $t\in[0,T]$ and all $\gamma,\delta\in\{1,\dots,l\}$, the functions $\mathbf{1}_{\Omega_0}\,\mathfrak{s}^\gamma_{\min(t,\tau)}$, $\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(t,\tau)}\cdot Z_t\,\mathfrak{s}_{\min(t,\tau)}$, and $\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(t,\tau)}\cdot Z_{\min(t,\tau)}\,\mathfrak{s}_{\min(t,\tau)}$ are bounded random variables; the expectation $\mathbb{E}[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0]$ is finite, the random variable being bounded ($|\mathfrak{s}_0|\le2\sqrt{N}$ everywhere and $Z_0$ a fixed matrix); the expectations

$$\mathbb{E}\big[\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(s,\tau)}\cdot\dot{z}(s)\,\mathfrak{s}_{\min(s,\tau)}\big],\qquad \mathbb{E}\big[\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau\}}\,\mathfrak{s}_s\cdot\dot{z}(s)\,\mathfrak{s}_s\big],\qquad \mathbb{E}\big[\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau\}}\,\mathfrak{s}_s\cdot Z_sg_s\big],\qquad \mathbb{E}\big[\mathbf{1}_{\{s<\tau\}}\mathbf{1}_{\Omega_0}\,\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)\big]$$

are finite for every $s\in[0,T]$, and as functions of $s$ they are bounded and measurable on $[0,T]$ with the trace Borel $\sigma$-algebra, so the Lebesgue integrals below exist — the first summand of the integrand of part (c) being, by linearity, the sum of the second of these expectations and twice the third.

(b) (Stopped evolution, running weight.) For every $t\in[0,T]$,

$$\mathbb{E}\big[\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(t,\tau)}\cdot Z_t\,\mathfrak{s}_{\min(t,\tau)}\big]=\mathbb{E}\big[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0\big]+\int_{[0,t]}\Big(\mathbb{E}\big[\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(s,\tau)}\cdot\dot{z}(s)\,\mathfrak{s}_{\min(s,\tau)}\big]+2\,\mathbb{E}\big[\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau\}}\,\mathfrak{s}_s\cdot Z_sg_s\big]+\sum_{\gamma,\delta=1}^{l}Z^{\gamma\delta}_s\,\mathbb{E}\big[\mathbf{1}_{\{s<\tau\}}\mathbf{1}_{\Omega_0}\,\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)\big]\Big)\,ds .$$

(c) (Fully stopped evolution.) For every $t\in[0,T]$,

$$\mathbb{E}\big[\mathbf{1}_{\Omega_0}\,\mathfrak{s}_{\min(t,\tau)}\cdot Z_{\min(t,\tau)}\,\mathfrak{s}_{\min(t,\tau)}\big]=\mathbb{E}\big[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0\big]+\int_{[0,t]}\Big(\mathbb{E}\big[\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau\}}\big(\mathfrak{s}_s\cdot\dot{z}(s)\,\mathfrak{s}_s+2\,\mathfrak{s}_s\cdot Z_sg_s\big)\big]+\sum_{\gamma,\delta=1}^{l}Z^{\gamma\delta}_s\,\mathbb{E}\big[\mathbf{1}_{\{s<\tau\}}\mathbf{1}_{\Omega_0}\,\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)\big]\Big)\,ds .$$