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

Preliminaries. For each $j$, the space $(\Omega,\mathcal{F},P)$ with initial states $\varsigma_j$ and the common clocks satisfies conditions 1--3 of N-Agent Driving System by hypothesis, and condition 4 as well: the required product identities involve one event from $\sigma(\varsigma_j^1,\dots,\varsigma_j^N)$ and one from each of finitely many clock $\sigma$-algebras, and since $\sigma(\varsigma_j^1,\dots,\varsigma_j^N)\subseteq\mathcal{S}_0$ each such identity is an instance of the hypothesis that $\mathcal{S}_0$ together with the clock $\sigma$-algebras forms an independent family. Hence clause (vi) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics applies to the given solution for $h_j$ on this driving system, at time $r$ with the caps $(c_a)$: writing $\mathcal{H}_j$ for the $\sigma$-algebra generated by $\varsigma_j^1,\dots,\varsigma_j^N$ together with the capped clock variables, there is $\hat{C}_j\in\mathcal{H}_j$ with $P(C_j\triangle\hat{C}_j)=0$, and for every $F\in\mathcal{F}^{\mathrm{sys},j}_r$ there is $H\in\mathcal{H}_j$ with $P\bigl((F\cap C_j)\triangle(H\cap C_j)\bigr)=0$. Note $\mathcal{H}_j\subseteq\mathcal{H}$: every generator of $\mathcal{H}_j$ is either a capped clock variable or a $\varsigma_j^i$, which is $\mathcal{S}_0$-measurable.

Claim 1. First, $C=\bigcap_jC_j$ agrees mod null with $\hat{C}:=\bigcap_j\hat{C}_j\in\mathcal{H}$, since $C\triangle\hat{C}\subseteq\bigcup_j(C_j\triangle\hat{C}_j)$. Let $\mathcal{A}=\{F\in\mathcal{F}:\exists H\in\mathcal{H}\ \text{with}\ P((F\cap C)\triangle(H\cap C))=0\}$. Then $\Omega\in\mathcal{A}$ (take $H=\Omega$). If $F\in\mathcal{A}$ with witness $H$, then $F^{c}\in\mathcal{A}$ with witness $H^{c}$: within $C$, membership of $F\triangle H$ and of $F^{c}\triangle H^{c}$ coincide, so $(F^{c}\cap C)\triangle(H^{c}\cap C)=(F\cap C)\triangle(H\cap C)$. If $F_n\in\mathcal{A}$ with witnesses $H_n$, then $\bigcup_nF_n\in\mathcal{A}$ with witness $\bigcup_nH_n$, since $\bigl(\bigcup_nF_n\cap C\bigr)\triangle\bigl(\bigcup_nH_n\cap C\bigr)\subseteq\bigcup_n\bigl((F_n\cap C)\triangle(H_n\cap C)\bigr)$. So $\mathcal{A}$ is a $\sigma$-algebra. For $F\in\mathcal{F}^{\mathrm{sys},j}_r$ with clause-(vi) witness $H\in\mathcal{H}_j\subseteq\mathcal{H}$: $(F\cap C)\triangle(H\cap C)=\bigl((F\cap C_j)\triangle(H\cap C_j)\bigr)\cap C$, which is null. Hence $\mathcal{F}^{\mathrm{sys},j}_r\subseteq\mathcal{A}$ for every $j$, and therefore $\mathcal{F}^{\mathrm{sys},1}_r\vee\dots\vee\mathcal{F}^{\mathrm{sys},J}_r\subseteq\mathcal{A}$.

Claim 2. For an indicator $Z=\mathbf{1}_F$ with $F$ in the join, claim 1 provides $H$ with $\mathbf{1}_F\mathbf{1}_C=\mathbf{1}_H\mathbf{1}_C$ almost surely; take $\hat{Z}=\mathbf{1}_H$. For a nonnegative simple $Z=\sum_{k\le K}z_k\mathbf{1}_{F_k}$ take $\hat{Z}=\sum_{k\le K}z_k\mathbf{1}_{H_k}$, the finitely many null events uniting. For general $Z:\Omega\to[0,\infty]$ measurable with respect to the join, let $Z_n=2^{-n}\lfloor2^{n}(Z\wedge2^{n})\rfloor$, a nondecreasing sequence of nonnegative simple functions, measurable with respect to the join, with $Z_n\to Z$ pointwise; let $\hat{Z}_n$ be witnesses from the simple case and set $\hat{Z}=\limsup_n\hat{Z}_n$, an $\mathcal{H}$-measurable $[0,\infty]$-valued map (countable suprema and infima of measurable maps are measurable). Off the union of the countably many null events, on $C$ one has $\hat{Z}_n=Z_n$ for every $n$, hence $\hat{Z}=\lim_nZ_n=Z$ there; that is, $Z\mathbf{1}_C=\hat{Z}\mathbf{1}_C$ almost surely.

Claim 3. Fix a clock label $a$ and write $\mathcal{J}=\sigma(Y^{a}_{c_a+s}-Y^{a}_{c_a}:s\ge0)$. Step 1. $Y^{a}$ has independent increments (clause 2 of its definition as a homogeneous Poisson process) and $Y^{a}_0=0$ everywhere (property 1 of Counting Path and Its Jump Times). By part (b) of Grouping Independence and the Fresh-Start Sigma-Algebra of an Independent-Increment Process, $\mathcal{J}$ is independent of the natural-filtration $\sigma$-algebra $\mathcal{F}^{Y^{a}}_{c_a}=\sigma(Y^{a}_u:0\le u\le c_a)$. Step 2. Consider the $\sigma$-algebra $\mathcal{H}^{+}_{a}=\sigma\bigl(\mathcal{S}_0\cup\bigcup_{b\neq a}\sigma(Y^{b}_u:u\ge0)\cup\mathcal{F}^{Y^{a}}_{c_a}\bigr)$ of the statement, the union over the clock labels $b\neq a$. Fix $J_0\in\mathcal{J}$ and define the finite measures $\nu_1(F)=P(J_0\cap F)$ and $\nu_2(F)=P(J_0)P(F)$ on $\mathcal{H}^{+}_{a}$. Let $\mathcal{P}$ be the collection of sets $S\cap E_{b_1}\cap\dots\cap E_{b_k}\cap E$ with $k\ge0$, $S\in\mathcal{S}_0$, distinct clock labels $b_1,\dots,b_k$ all $\neq a$, $E_{b_i}\in\sigma(Y^{b_i}_u:u\ge0)$, and $E\in\mathcal{F}^{Y^{a}}_{c_a}$. Then $\mathcal{P}$ is a $\pi$-system (intersect componentwise, intersecting events of coincident labels within their $\sigma$-algebra), contains $\Omega$, and generates $\mathcal{H}^{+}_{a}$. For a member of $\mathcal{P}$: $J_0\cap E\in\sigma(Y^{a}_u:u\ge0)$, so the hypothesis independence of the family consisting of $\mathcal{S}_0$ and the clock $\sigma$-algebras gives $P\bigl(S\cap E_{b_1}\cap\dots\cap E_{b_k}\cap E\cap J_0\bigr)=P(S)\,P(E_{b_1})\cdots P(E_{b_k})\,P(E\cap J_0)=P(S)\,P(E_{b_1})\cdots P(E_{b_k})\,P(E)\,P(J_0),$ the last step by Step 1; and the same family independence gives $P(S\cap E_{b_1}\cap\dots\cap E_{b_k}\cap E)=P(S)P(E_{b_1})\cdots P(E_{b_k})P(E)$. Hence $\nu_1=\nu_2$ on $\mathcal{P}$, with equal total masses $\nu_1(\Omega)=P(J_0)=\nu_2(\Omega)$. By claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, $\nu_1=\nu_2$ on $\sigma(\mathcal{P})=\mathcal{H}^{+}_{a}$. Since $J_0\in\mathcal{J}$ was arbitrary, $\mathcal{J}$ is independent of $\mathcal{H}^{+}_{a}$. The final clause of the statement follows by restriction: $\mathcal{H}\subseteq\mathcal{H}^{+}_{a}$ (every generator of $\mathcal{H}$ is $\mathcal{S}_0$-measurable, a capped variable of a clock $b\neq a$, hence $\sigma(Y^{b}_u:u\ge0)$-measurable, or a capped variable of $a$, hence $\mathcal{F}^{Y^{a}}_{c_a}$-measurable), and increments of a clock $b\neq a$ beyond arbitrary levels are $\sigma(Y^{b}_u:u\ge0)$-measurable, so every $\sigma$-algebra named there is a sub-$\sigma$-algebra of $\mathcal{H}^{+}_{a}$, and independence restricts to sub-$\sigma$-algebras.