The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time

Adopt the setting and notation of the controlled $N$-agent dynamics: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$; a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$, assumed convex and compact for the topology determined by the Euclidean distance (as required by the realized-control lemma, which is used below); a transition-rate family $\beta$ with rate bound $B$; an observation-rate family $\tilde\beta$ with $\tilde{l}$ channels and rate bound $\tilde{B}$; a real number $T>0$; an $N$-agent driving system $(\Omega,\mathcal{F},P)$; and an $\mathcal{A}$-valued observation-driven control policy $h$ with horizon $T$. Let a solution on $[0,T]$ for $h$ be given, with regular event $\Omega_0$, observation processes $\Upsilon^{\upsilon}$, observation counters $\tilde{N}^{i,\upsilon}$, observation total $\tilde{c}$, observation-event count $K_t=\tilde{c}_t$, observation event times $\tau_1<\dots<\tau_{K_T}$ and channels $\upsilon_1,\dots,\upsilon_{K_T}$ as in condition 5 of Solution of the Controlled N-Agent Dynamics, observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$, and observation record $W$; the random variables $\tau_j$ and $\upsilon_j$ ($j\ge1$) are those furnished by claim 1 of the realized-control lemma, so that $\tilde{c}_t(\omega)\ge j$ if and only if $\tau_j(\omega)\le t$ for all $\omega\in\Omega$, $j\ge1$ and $t\in[0,T]$, and at every $\omega\in\Omega_0$ the numbers $\tau_j(\omega)$ and $\upsilon_j(\omega)$ with $1\le j\le K_T(\omega)$ are the event times and channels of condition 5.

For $\upsilon\in\{1,\dots,\tilde{l}\}$ and $u\in[0,T]$ put $\tilde{c}^{\upsilon}_u=\sum_{i=1}^{N}\tilde{N}^{i,\upsilon}_u$, the channel subtotal, so that $\tilde{c}^{\upsilon}_u=N\,\Upsilon^{\upsilon}_u$ by condition 4 of Solution of the Controlled N-Agent Dynamics at every $\omega\in\Omega_0$, and $\tilde{c}_u=\sum_{\upsilon=1}^{\tilde{l}}\tilde{c}^{\upsilon}_u$ by condition 3 there.

Fix a real number $s$ with $0<s\le T$. Let $(\mathbf{R}(s,\tilde{l}),\mathcal{R}(s,\tilde{l}))$ be the observation record space with horizon $s$ and $\tilde{l}$ channels, and let $\pi_s:\mathbf{R}(T,\tilde{l})\to\mathbf{R}(s,\tilde{l})$ be the prefix map when $s<T$ and the identity map when $s=T$. Put $W^{(s)}=\pi_s\circ W$. When $s<T$, let $h^{(s)}$ be the truncated policy and let the restricted solution on $[0,s]$ be the one furnished by claim 2 of that lemma, whose observation record is $W^{(s)}$ and whose observation filtration satisfies $\mathcal{G}^{(s)}_u=\mathcal{G}_u$ for $u\in[0,s]$; assume that reconstruction data are fixed for the driving system and the policy $h^{(s)}$ with horizon $s$ (and, when $s=T$, for the driving system and $h$ with horizon $T$).

Write $\mathcal{N}=\{A\in\mathcal{F}:P(A)=0\}$, $\sigma(W^{(s)})=\{(W^{(s)})^{-1}(A):A\in\mathcal{R}(s,\tilde{l})\}$, $\mathbf{1}_{A}$ for the indicator of a set $A$, and $\#E$ for the number of elements of a finite set $E$. Measurability of real-valued maps is with respect to the named $\sigma$-algebra and the Borel $\sigma$-algebra of the real line. Two notational cautions: the letter $\sigma$ occurs both as a state label and in the state processes $\sigma^{i}$ of Solution of the Controlled N-Agent Dynamics and as the symbol for a generated $\sigma$-algebra, the latter always applied to a map and written $\sigma(\cdot)$; and the family $\mathcal{N}$ of null events is unrelated to the agent count $N$. Then the following hold.

1. (The record up to time $s$ is observable.) $W^{(s)}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}(s,\tilde{l})$, and $\sigma(W^{(s)})\subseteq\mathcal{G}_s$.

2. (Counting identities.) For every $\omega\in\Omega_0$ and every $u\in[0,T]$,

$$\tilde{c}_u(\omega)=\#\{j\ge1:\tau_j(\omega)\le u\},\qquad \tilde{c}^{\upsilon}_u(\omega)=\#\{j\ge1:\tau_j(\omega)\le u\text{ and }\upsilon_j(\omega)=\upsilon\}\quad(1\le\upsilon\le\tilde{l}),$$

both sets being finite.

3. (The observation processes factor through the record.) For every $\upsilon\in\{1,\dots,\tilde{l}\}$ and every $u\in[0,s]$ there is a map $\chi^{\upsilon}_u:\mathbf{R}(s,\tilde{l})\to\mathbb{R}$, measurable with respect to $\mathcal{R}(s,\tilde{l})$, such that

$$\Upsilon^{\upsilon}_u(\omega)=\chi^{\upsilon}_u\bigl(W^{(s)}(\omega)\bigr)\qquad\text{for every }\omega\in\Omega_0 .$$

Explicitly, $\chi^{\upsilon}_u(k,\mathbf{t},v)=\frac{1}{N}\#\{j\in\{1,\dots,k\}:t_j\le u\text{ and }v_j=\upsilon\}$ for a record $(k,\mathbf{t},v)\in\mathbf{R}(s,\tilde{l})$.

4. (The observation filtration.) $\mathcal{G}_s$ is the $\sigma$-algebra generated by $\sigma(W^{(s)})\cup\mathcal{N}$. Consequently $\mathcal{G}_s$ is $W^{(s)}$-generated up to null sets in the sense of claim 1 of that lemma, and for every square-integrable random variable $X$ on $(\Omega,\mathcal{F},P)$,

$$\mathbb{E}\Bigl[\bigl(X-\mathbb{E}[X\mid\mathcal{G}_s]\bigr)^{2}\Bigr]=\mathbb{E}\Bigl[\bigl(X-\mathbb{E}[X\mid\sigma(W^{(s)})]\bigr)^{2}\Bigr].$$