The Control of the Controlled N-Agent Dynamics is Adapted to the Observation Filtration

Adopt the setting of the definition of a solution of the controlled $N$-agent dynamics on $[0,T]$: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, a nonempty subset $\mathcal{A}$ of Euclidean space $\mathbb{R}^m$, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ channels, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h=(h_k)_{k\ge0}$ which is $\mathcal{A}$-valued, and a solution with regular event $\Omega_0$, observation processes $\Upsilon^\upsilon$, observation total $\tilde{c}_t$, and control process $\alpha$. Let $(\mathcal{G}_t)_{t\in[0,T]}$ be the observation filtration of the solution definition, and call a random variable $\mathcal{G}_t$-measurable if its preimages of Borel sets lie in $\mathcal{G}_t$, as in the criterion of the definition of a measurable function. Then for every $t\in[0,T]$ and every $j\in\{1,\dots,m\}$:

1. (Adaptedness.) $\alpha^j_t$ is $\mathcal{G}_t$-measurable. Moreover there is a $\mathcal{G}_t$-measurable random variable $\alpha'^{\,j}_t$ on $(\Omega,\mathcal{F},P)$ with $\alpha^j_t(\omega)=\alpha'^{\,j}_t(\omega)$ at every $\omega\in\Omega_0$; in particular $\alpha^j_t$ is almost surely equal to $\alpha'^{\,j}_t$.

2. (Control fluctuation.) Consequently, if moreover $(S,A)$ is a mean-field trajectory pair for $\beta$ with horizon $T$, so that $A_t\in\mathcal{A}$ for every $t\in[0,T]$, and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$ is the control fluctuation process of the solution about $(S,A)$, then each component $\mathfrak{a}^j_t$ is $\mathcal{G}_t$-measurable; it equals the $\mathcal{G}_t$-measurable random variable $\sqrt{N}(\alpha'^{\,j}_t-A^j_t)$ at every $\omega\in\Omega_0$ and hence is almost surely equal to it; and if $\mathfrak{a}^j_t$ is square-integrable, then so is that variable, with the same mean-square norm, by the transfer lemma for almost sure equality.