Proof of Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon

Throughout, "the recursion" is the aggregate recursion of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution for $(\mathsf{P}(\omega),a,x_0)$, with the notation of that theorem ($\theta_k$, $x^{(k)}$, $\kappa^{c}_k$, $\mathsf{C}^{c,(k)}$, $\lambda^{c}_k$, $h^{c}_k$, $\mathcal{J}_k$, $K$), and $c=(\sigma,\gamma)$ always denotes a transition label; there are $l(l-1)$ labels. We verify the data and hypotheses of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property one by one (claim 1), and then read off claim 2.

(D1). Every path of $\mathsf{P}^{c}$ is a counting path by property 1 of Poisson Clock with a Horizon.

(D2). For fixed $x\in\mathbb{G}_N$ the map $u\mapsto\beta(\sigma,\gamma,x,a_u)$ is measurable on $[0,T]$: the control path $a$ has measurable components, and $\alpha\mapsto\beta(\sigma,\gamma,x,\alpha)$ is sequentially continuous on $\mathcal{A}$ by condition 2 of Transition-Rate Family, so Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable applies (this is the argument recorded in Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks). Multiplying by the constant $Nx^{\sigma}$ keeps measurability (Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and $0\le g^{c}(u,x)\le NB$ because $0\le x^{\sigma}\le1$ and $0\le\beta\le B$ (condition 1 of Transition-Rate Family). Thus $\Lambda=NB$ serves. The maps $\phi^{c}$ take values in $\mathbb{G}_N$ by construction.

(D3). The recursion filtration is nested by definition. Let $\iota:\mathbb{R}^l\to\mathbb{G}_N$ be the map with $\iota(x)=x$ for $x\in\mathbb{G}_N$ and $\iota(x)=x_0$ otherwise; since $\mathbb{G}_N$ is finite, $\iota^{-1}(\{x\})$ is a Borel subset of $\mathbb{R}^l$ (a singleton, or the complement of a finite set united with a singleton) for every $x\in\mathbb{G}_N$. As $\bar{\Sigma}_t=\iota(\Sigma_t)$ and each coordinate $\Sigma^{\gamma}_t$ is a generator of $\mathfrak{F}_t$, the event $\{\bar{\Sigma}_t=x\}=\{\Sigma_t\in\iota^{-1}(\{x\})\}$ belongs to $\mathfrak{F}_t$: $\{\Sigma_t=x\}=\bigcap_{\gamma}\{\Sigma^{\gamma}_t=x^{\gamma}\}$ for each $x$, and $\{\Sigma_t\notin\mathbb{G}_N\}$ is the complement of the finite union of the events $\{\Sigma_t=x\}$, $x\in\mathbb{G}_N$. For the joint measurability, claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution with a one-point parameter space gives that $(u,\omega)\mapsto\Sigma^{\gamma}_u(\omega)$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable for each $\gamma$, so $\{(u,\omega):\Sigma_u(\omega)=x\}$ is product-measurable for each $x\in\mathbb{R}^l$, hence so is $\{(u,\omega):\bar{\Sigma}_u(\omega)=x\}$ for $x\in\mathbb{G}_N$ (equal to $\{\Sigma_u=x\}$ for $x\neq x_0$, and to $\{\Sigma_u=x_0\}\cup\{\Sigma_u\notin\mathbb{G}_N\}$ for $x=x_0$), and multiplying its indicator by $\mathbf{1}_{\Omega_0}(\omega)$ (a rectangle indicator) preserves product measurability.

(D4). Each $\mathsf{C}^{c}_t$ is a random variable with values in $[0,NBt]$ (claim 4 and claim 1 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, as in Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels). Let $\omega\in\Omega_0$. By claim 2 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, the data being conflict-free, the recursion path is the unique open-loop aggregate solution for $(\mathsf{P}(\omega),a,x_0)$ and the recursion consumed times are the consumed clock times of that solution, which Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks defines as $\int_{[0,t]}N\Sigma^{\sigma}_u\beta(\sigma,\gamma,\Sigma_u,a_u)\,du$; since $\Sigma=\bar{\Sigma}$ on $\Omega_0$ (the solution takes values in $\mathbb{G}_N$), this is $\int_{[0,t]}g^{c}(u,\bar{\Sigma}_u)\,du$. Hence (D4) holds with $\mathcal{T}^{c}_t=\mathsf{C}^{c}_t$, and the counters of the generic lemma are $\mathsf{P}^{c}(\mathsf{C}^{c}_t)=\mathsf{N}^{c}_t$.

(H1). Fix $\omega\in\Omega_0$, so that the recursion stops at $K$ with $\theta_K=T$ and $x^{(k)}\in\mathbb{G}_N$ for all $k\le K$ (the recursion stops at the first index where a point leaves $\mathbb{G}_N$, so conflict-freeness forces all points to lie in $\mathbb{G}_N$), and $\bar{\Sigma}=\Sigma$. The path $t\mapsto\Sigma_t$ equals $x^{(k)}$ on $[\theta_k,\theta_{k+1})$ for $k<K$ and $x^{(K)}$ at $\theta_K=T$, with $0=\theta_0<\theta_1<\dots<\theta_K=T$ (claim 1): this is the piecewise-constant right-continuous structure required in (H1), with the times $\theta_1,\dots,\theta_K$ (the count of (H1) coincides with the stopping index $K$ of the recursion). Counters between and at recursion times. Fix $k<K$ and a label $c$, and let $j=\mathsf{P}^{c}(\kappa^{c}_k)+1$, so that $\lambda^{c}_k=\tau_j(\mathsf{P}^{c}(\omega))$, the greatest lower bound of the times at which the counting path $\mathsf{P}^{c}(\omega)$ has value at least $j$ ($+\infty$ if there are none). This path is at least $j-1$ on $[\kappa^{c}_k,\infty)$ by monotonicity, and below $j$ on $[0,\lambda^{c}_k)$ by the definition of $\lambda^{c}_k$, so it equals $j-1$ on $[\kappa^{c}_k,\lambda^{c}_k)$; and when $\lambda^{c}_k<+\infty$ it equals $j$ at $\lambda^{c}_k$ (its value there is at least $j$ by right-continuity, and at most $(j-1)+1$ since its left limit there is at most $j-1$ and jumps are of size at most $1$; Counting Path and Its Jump Times). Only the case $\lambda^{c}_k<+\infty$ is used below, since it is forced for $c\in\mathcal{J}_k$ by $\mathsf{C}^{c,(k)}_{\theta_{k+1}}\ge\lambda^{c}_k$. For $t\in[\theta_k,\theta_{k+1})$ one has $\kappa^{c}_k\le\mathsf{C}^{c,(k)}_t<\lambda^{c}_k$ (the first inequality by monotonicity of $\mathsf{C}^{c,(k)}$ from its value $\kappa^{c}_k$ at $\theta_k$; the second because $\mathsf{C}^{c,(k)}_t\ge\lambda^{c}_k$ would give $h^{c}_k\le t<\theta_{k+1}\le h^{c}_k$), so $\mathsf{N}^{c}_t=\mathsf{P}^{c}(\mathsf{C}^{c,(k)}_t)=j-1$ is constant there. At $t=\theta_{k+1}$, $\mathsf{N}^{c}_{\theta_{k+1}}=\mathsf{P}^{c}(\kappa^{c}_{k+1})$ with $\kappa^{c}_{k+1}=\mathsf{C}^{c,(k)}_{\theta_{k+1}}$: if $c\in\mathcal{J}_k$ then $\kappa^{c}_{k+1}=\lambda^{c}_k$ (claim 1) and $\mathsf{N}^{c}_{\theta_{k+1}}=j$, while if $c\notin\mathcal{J}_k$ then $\kappa^{c}_k\le\kappa^{c}_{k+1}<\lambda^{c}_k$ and $\mathsf{N}^{c}_{\theta_{k+1}}=j-1$. Hence every counter is constant on each $[\theta_k,\theta_{k+1})$, and at $\theta_{k+1}$ exactly the counters with labels in $\mathcal{J}_k$ jump, each by $1$. The jump rules. Let $t\in(0,T]$ and let $\mathsf{N}^{\mathrm{tot}}_t=\sum_c\mathsf{N}^{c}_t$ be the grand total (written $N_t$ in Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property; here $N$ is the number of agents). If $t$ is not one of $\theta_1,\dots,\theta_K$, then $t$ lies in the interior of some $[\theta_k,\theta_{k+1})$, where all counters and $\Sigma$ are constant, so $\mathsf{N}^{\mathrm{tot}}_t=\mathsf{N}^{\mathrm{tot}}_{t-}$ and $\Sigma_t=\Sigma_{t-}$. If $t=\theta_{k+1}$ with $k<K$: the left limits are the constant values on $[\theta_k,\theta_{k+1})$, so $\mathsf{N}^{\mathrm{tot}}_t-\mathsf{N}^{\mathrm{tot}}_{t-}=|\mathcal{J}_k|$ and $\Sigma_t-\Sigma_{t-}=x^{(k+1)}-x^{(k)}=\frac1N\sum_{c\in\mathcal{J}_k}v_c$. If $\mathcal{J}_k=\emptyset$ this gives $\mathsf{N}^{\mathrm{tot}}_t=\mathsf{N}^{\mathrm{tot}}_{t-}$ and $\Sigma_t=\Sigma_{t-}$; if $\mathcal{J}_k=\{c\}$ then $\mathsf{N}^{\mathrm{tot}}_t=\mathsf{N}^{\mathrm{tot}}_{t-}+1$, $c$ is the unique label whose counter jumps at $t$, and $\Sigma_t=\Sigma_{t-}+\frac1Nv_c$, which lies in $\mathbb{G}_N$ (it is $x^{(k+1)}$), so $\Sigma_t=\phi^{c}(\Sigma_{t-})$; if $|\mathcal{J}_k|\ge2$ nothing is required. This is (H1).

(H2). Fix $r\in[0,T]$. We apply Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound with the clock labels $\mathsf{A}=$ the transition labels, the initial data $\mathcal{I}=\{\emptyset,\Omega\}$, the clocks $Y^{c}=\mathsf{P}^{c}$ (Poisson clocks with horizon $R$; the family consisting of the trivial $\sigma$-algebra $\mathcal{I}$ and the $\sigma(\mathsf{P}^{c}_u:u\ge0)$ is independent, since adjoining $\{\emptyset,\Omega\}$ to an independent family keeps every product formula true), the past $\mathfrak{F}=\mathfrak{F}_r$, the consumed levels $\mathcal{T}^{c}=\mathsf{C}^{c}_r$ and $\bar{c}=NBr$: these levels are $\mathfrak{F}_r$-measurable with $0\le\mathsf{C}^{c}_r\le NBr<R$, and they satisfy the clock-reading bound by claim 2 of Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels. Hence the residual clocks $\hat{Y}^{c,r}_u=\mathsf{P}^{c}_{\mathsf{C}^{c}_r+u}-\mathsf{P}^{c}_{\mathsf{C}^{c}_r}$ consist of random variables, have independent Poisson increments (with the parameters of (H2)) over every finite time set in $[0,R^-)$ with $R^-=R-NBr$, and the family consisting of $\mathfrak{F}_r$ and the $\sigma$-algebras $\sigma(\hat{Y}^{c,r}_u:0\le u<R^-)$ is independent. Since $\varrho_r=\Lambda(T-r)=NB(T-r)<R-NBr=R^-$ (because $NBT<R$), every finite time set in $[0,\varrho_r]$ lies in $[0,R^-)$ and $\sigma(\hat{Y}^{c,r}_u:0\le u\le\varrho_r)\subseteq\sigma(\hat{Y}^{c,r}_u:0\le u<R^-)$; independence of a family of $\sigma$-algebras passes to sub-$\sigma$-algebras (Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras). This is (H2), and claim 1 is proved.

Claim 2. By claim 1, claim (a) of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property applies to $X=\bar{\Sigma}$ and yields the displayed identity, the integrand there being $\mathbf{1}_{\Omega_0}\sum_cg^{c}(u,\bar{\Sigma}_u)(F(\phi^{c}(\bar{\Sigma}_u))-F(\bar{\Sigma}_u))$; when $\bar{\Sigma}_u+\frac1Nv_c\notin\mathbb{G}_N$ one has $\phi^{c}(\bar{\Sigma}_u)=\bar{\Sigma}_u$ and the term vanishes, which matches the stated reading (and, for $x=\bar{\Sigma}_u\in\mathbb{G}_N$ (which holds at every $\omega$, by the definition of $\bar{\Sigma}$), the point $x+\frac1Nv_c=x+\frac1N(\delta_\gamma-\delta_\sigma)$ fails to lie in $\mathbb{G}_N$ exactly when $x^{\sigma}=0$, its coordinates otherwise being nonnegative multiples of $\frac1N$ summing to $1$). Claim (b) of the generic lemma gives that $(\mu^{D}_u)_{u\in[r,T]}$ solves the forward equation on $[r,T]$ for the rates $q_u(x,y)=\sum_{c:\phi^{c}(x)=y}g^{c}(u,x)$ with rate bound $|\mathsf{A}|\Lambda=l(l-1)NB$. Finally, for $x\neq y$: if $y=x+\frac1Nv_c$ for some label $c$ then $c$ is unique ($v_c=\delta_\gamma-\delta_\sigma$ determines $(\sigma,\gamma)$), $\phi^{c}(x)=y$, no other label $c'$ has $\phi^{c'}(x)=y$ (as $\phi^{c'}(x)\in\{x,x+\frac1Nv_{c'}\}$), and $q_u(x,y)=g^{c}(u,x)=Nx^{\sigma}\beta(\sigma,\gamma,x,a_u)$; otherwise no label has $\phi^{c}(x)=y$ and $q_u(x,y)=0$, as stated.