Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics

Let $N$, $l$, $\tilde{l}$, $m$ be natural numbers with $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, let $\beta$ be a transition-rate family on $l$ states with control dimension $m$ and rate bound $B$, let $\tilde{\beta}$ be an observation-rate family with $\tilde{l}$ channels and rate bound $\tilde{B}$, and let $T>0$ be a real number. Let $(\Omega,\mathcal{F},P)$ be a probability space carrying transition clocks $Y^{i,\sigma\gamma}$ and observation clocks $\tilde{Y}^{i,\upsilon}$ as in conditions 2 and 3 of N-Agent Driving System, and let $\mathcal{S}_0\subseteq\mathcal{F}$ be a $\sigma$-algebra such that the finite family of $\sigma$-algebras consisting of $\mathcal{S}_0$ together with the $\sigma$-algebra generated by the variables of each single clock is independent. As in Compensated Counters of the Controlled N-Agent Dynamics are Square-Integrable Martingales, a clock label $a$ is either a pair $(i,\sigma\gamma)$ with $\sigma\neq\gamma$ or a pair $(i,\upsilon)$; write $Y^{a}$ for the corresponding clock and $B_a$ for $B$ when $a$ is a transition label and for $\tilde{B}$ when $a$ is an observation label (the bounds $B_a$ are used by results adopting this setting).

Let $J\ge1$ be a natural number and, for each $j\in\{1,\dots,J\}$, let $\varsigma_j=(\varsigma_j^1,\dots,\varsigma_j^N)$ be $\mathcal{S}_0$-measurable random variables with values in $\{1,\dots,l\}$. Then $(\Omega,\mathcal{F},P)$ with the initial states $\varsigma_j$ and the common clocks is an $N$-agent driving system (condition 4 there holding since $\sigma(\varsigma_j^1,\dots,\varsigma_j^N)\subseteq\mathcal{S}_0$). For each $j$ let $h_j$ be an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, and let a solution of the controlled $N$-agent dynamics on $[0,T]$ for $h_j$ on the $j$-th driving system be given, with consumed clock times $A_j^{a}$ (indexed by the clock labels $a$) and system filtration $(\mathcal{F}^{\mathrm{sys},j}_t)_{t\in[0,T]}$.

Fix $r\in[0,T]$ and a nonnegative real $c_a$ for each clock label $a$. Let $\mathcal{H}$ be the $\sigma$-algebra generated by $\mathcal{S}_0$ together with the clock variables $Y^{a}_u$ for all clock labels $a$ and all real $u$ with $0\le u\le c_a$; let $C_j$ be the event that $A_j^{a}(r)\le c_a$ for every clock label $a$; and let $C=C_1\cap\dots\cap C_J$.

1. (Embedding of events) The event $C$ agrees up to an event of probability zero with an event of $\mathcal{H}$; and for every $F$ in the $\sigma$-algebra $\mathcal{F}^{\mathrm{sys},1}_r\vee\dots\vee\mathcal{F}^{\mathrm{sys},J}_r$ generated by the union of the system $\sigma$-algebras at time $r$, there is $H\in\mathcal{H}$ such that the symmetric difference of $F\cap C$ and $H\cap C$ is an event of probability zero.

2. (Embedding of random variables) For every $\mathcal{F}^{\mathrm{sys},1}_r\vee\dots\vee\mathcal{F}^{\mathrm{sys},J}_r$-measurable $Z:\Omega\to[0,\infty]$ there is an $\mathcal{H}$-measurable $\hat{Z}:\Omega\to[0,\infty]$ with $Z\,\mathbf{1}_C=\hat{Z}\,\mathbf{1}_C$ almost surely.

3. (Fresh clock beyond its cap) For every clock label $a$, the $\sigma$-algebra $\sigma(Y^{a}_{c_a+s}-Y^{a}_{c_a}:\ s\ge0)$ generated by the increments of $Y^{a}$ beyond $c_a$ is independent of the $\sigma$-algebra $\mathcal{H}^{+}_{a}=\sigma\Bigl(\mathcal{S}_0\ \cup\ \bigcup_{b\neq a}\sigma\bigl(Y^{b}_u:u\ge0\bigr)\ \cup\ \sigma\bigl(Y^{a}_u:0\le u\le c_a\bigr)\Bigr),$ the union over all clock labels $b\neq a$; in particular it is independent of $\mathcal{H}$, and of every $\sigma$-algebra generated by $\mathcal{H}$ together with increments of clocks other than $a$ beyond arbitrary levels.