Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics

Adopt the setting and notation of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ in Euclidean space, a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$, an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$ with initial states $\varsigma^i_0$, transition clocks $Y^{i,\sigma\gamma}$ and observation clocks $\tilde{Y}^{i,\upsilon}$, an $\mathcal{A}$-valued observation-driven control policy $h$, the observation record space $\mathbf{R}$ with record $\sigma$-algebra $\mathcal{R}$, the record-frozen control path $a^r$ and record-frozen policy $\hat{h}^r$ at $r\in\mathbf{R}$, the $\sigma$-algebra $\mathcal{T}$, the reconstructed states $\sigma^{r,i}_s$, occupation indicators $\eta^{r,i,\gamma}_s$ and empirical measures $\Sigma^r_s$, the good set $G$, and, for each $r\in\mathbf{R}$, the event $\Omega^r$ and the solution of the controlled $N$-agent dynamics for the policy $\hat{h}^r$ furnished by clause (d) of that lemma, whose state processes are the $\sigma^{r,i}$, whose control process is $s\mapsto a^r(s)$, and whose regular event is $\Omega^r$. For this solution write $\mathcal{T}^{i,\sigma\gamma}_t$ for its consumed transition-clock times, $N^{i,\sigma\gamma}_t=Y^{i,\sigma\gamma}_{\mathcal{T}^{i,\sigma\gamma}_t}$ for its transition counters, and $(\mathcal{F}^{\mathrm{sys},r}_t)_{t\in[0,T]}$ for its system filtration, all as in Solution of the Controlled N-Agent Dynamics (the dependence on the record $r$ is suppressed in the first two symbols; from the second paragraph on, $r$ is fixed). Let $\mathbb{G}_N$ be the aggregate lattice (a nonempty finite set), let $c=(\sigma,\gamma)$ range over the transition labels with the vectors $v_c=\delta_\gamma-\delta_\sigma$ of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks, write $\mathbb{E}$ for the expectation under $P$, $\mathcal{B}_{[0,T]}$ for the trace Borel $\sigma$-algebra on $[0,T]$, integrals over compact intervals for the Lebesgue integral over the compact interval (read as $0$ over a degenerate interval), $\mathbf{1}_S$ for the indicator of a set $S$, and $\mathbf{1}\{\cdots\}$ for the indicator of the condition in the braces.

Fix $r\in\mathbf{R}$. Let $E=\{1,\dots,l\}^N$, a nonempty finite set with $l^N$ elements; for $x=(x^1,\dots,x^N)\in E$ let $\Sigma(x)\in\mathbb{R}^l$ be the point with coordinates $\Sigma(x)^\gamma=\frac{1}{N}\,\#\{i\in\{1,\dots,N\}:x^i=\gamma\}$, a point of $\mathbb{G}_N$. Put

both defined at every $\omega\in\Omega$. Let $\mathsf{A}$ be the set of pairs $e=(i,c)$ of an agent index $i\in\{1,\dots,N\}$ and a transition label $c=(\sigma,\gamma)$, a nonempty finite set with $Nl(l-1)$ elements (the clock labels, written $a$ in Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property, are written $e$ here, $a^r$ being reserved for the record-frozen control path), and for $e=(i,(\sigma,\gamma))\in\mathsf{A}$, $u\in[0,T]$ and $x\in E$ put

the last two formulas defining the point $\phi^{e}(x)\in E$ coordinatewise. Finally, for $u\in[0,T]$ and $y\neq y'$ in $\mathbb{G}_N$ put $q^r_u(y,y')=N\,y^{\sigma}\,\beta(\sigma,\gamma,y,a^r(u))$ if $y'=y+\frac{1}{N}v_c$ for the (then unique) transition label $c=(\sigma,\gamma)$, and $q^r_u(y,y')=0$ if $y'-y$ is not of the form $\frac1Nv_c$. Then:

1. (The reconstructed solution is a Poisson-clock jump system.) On $\Omega^r$ one has $\eta^{r,i,\gamma}_t=\mathbf{1}\{\sigma^{r,i}_t=\gamma\}$ for all $t\in[0,T]$, $i$ and $\gamma$, and $\hat{\Sigma}^r_t=\Sigma^r_t$ for all $t\in[0,T]$; moreover $a^r$ is a control path in the sense of that definition. The data consisting of $(\Omega,\mathcal{F},P)$, $T$, $\Lambda=B$, the state space $E$, the clock labels $\mathsf{A}$, the clocks $Y^{(i,(\sigma,\gamma))}=Y^{i,\sigma\gamma}$, the rate functions $g^{e}$, the transition maps $\phi^{e}$, the filtration $(\mathcal{F}^{\mathrm{sys},r}_t)_{t\in[0,T]}$, the event $\Omega_0=\Omega^r$, the state process $X$ and the consumed clock times $\mathcal{T}^{(i,(\sigma,\gamma))}_t=\mathcal{T}^{i,\sigma\gamma}_t$ satisfy (D1)--(D4), (H1) and (H2) of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property, and the counters of that lemma are the transition counters $N^{i,\sigma\gamma}_t$. Consequently conclusions (a) and (b) of that lemma hold for these data (the time parameter written $r$ in (H2) and in conclusions (a) and (b) of that lemma is here written $s$; the letter $r$ is reserved for the record).

2. (Forward equation on the aggregate lattice.) For every function $F:\mathbb{G}_N\to\mathbb{R}$, all $0\le s\le t\le T$ and every event $D\in\mathcal{F}^{\mathrm{sys},r}_s$,

where $\hat{\Sigma}^{r,\sigma}_u$ is the $\sigma$-th coordinate of $\hat{\Sigma}^r_u$, where $F(\hat{\Sigma}^r_u+\frac1Nv_c)$ is read as $F(\hat{\Sigma}^r_u)$ when $\hat{\Sigma}^r_u+\frac1Nv_c\notin\mathbb{G}_N$ (in which case $\hat{\Sigma}^{r,\sigma}_u=0$, so the term vanishes anyway), and where the inner integral is defined for every $\omega$ and is $\mathcal{F}$-measurable, its integrand being a bounded section of a $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable map. Moreover, for all $y\neq y'$ in $\mathbb{G}_N$ the map $u\mapsto q^r_u(y,y')$ is measurable on $[0,T]$ with respect to $\mathcal{B}_{[0,T]}$, $\sum_{y'\neq y}q^r_u(y,y')\le l(l-1)NB$ for all $u\in[0,T]$ and $y\in\mathbb{G}_N$, and for every $s\in[0,T)$ and $D\in\mathcal{F}^{\mathrm{sys},r}_s$ the family

(both sets in the braces being events) is a solution of the forward equation on $[s,T]$ for the rates $q^r$ in the sense of that lemma, applied with the rate bound there taken to be $l(l-1)NB$ and with its pairing $\nu(F)=\sum_{y\in\mathbb{G}_N}\nu(y)F(y)$.