Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records

Adopt the setting of the controlled $N$-agent dynamics with $N\ge1$ agents, $l\ge2$ states, $\tilde{l}\ge1$ observation channels, and control dimension $m\ge1$, with 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 $\tilde{Y}^{i,\upsilon}$, and an observation-driven control policy $h$ which is $\mathcal{A}$-valued. Let $\mathbf{R}=\mathbf{R}(T,\tilde{l})$ be the observation record space with record $\sigma$-algebra $\mathcal{R}$, for $r\in\mathbf{R}$ let $a^r$, $\hat{h}^r$, and $k_r$ be the record-frozen control path, the record-frozen policy, and the event count at $r$, and let $\mathcal{T}\subseteq\mathcal{F}$ be the $\sigma$-algebra generated by the initial states $\varsigma^1_0,\dots,\varsigma^N_0$ together with the transition-clock variables $Y^{i,\sigma\gamma}_u$ for all indices and all $u\ge0$ (the bare symbol $\mathcal{T}$ is reserved for this $\sigma$-algebra; the consumed clock times of condition 2 of Solution of the Controlled N-Agent Dynamics always carry superscripts, as in $\tilde{\mathcal{T}}^{i,\upsilon}$, and for any family of processes satisfying conditions 1--4 of that definition they are well defined and bounded by clause (vii)(b) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics).

Then there exist functions $\eta^{r,i,\gamma}_s(\omega)\in\{0,1\}$, states $\sigma^{r,i}_s(\omega)\in\{1,\dots,l\}$, and consumed times $\tilde{A}^{r,i,\upsilon}_s(\omega)\in[0,\tilde{B}T]$, defined for all $r\in\mathbf{R}$, $s\in[0,T]$, $\omega\in\Omega$, $i\in\{1,\dots,N\}$, $\gamma\in\{1,\dots,l\}$, and $\upsilon\in\{1,\dots,\tilde{l}\}$, together with a set $G\in\mathcal{R}\otimes\mathcal{T}$ (product $\sigma$-algebra) that contains $\mathbf{R}\times\Omega_G$ for some event $\Omega_G$ with $P(\Omega_G)=1$, with the following properties. Write $\Sigma^{r,\gamma}_s=\frac{1}{N}\sum_{i=1}^N\eta^{r,i,\gamma}_s$ and $\Sigma^r_s=(\Sigma^{r,1}_s,\dots,\Sigma^{r,l}_s)$.

(a) (Joint measurability) For all $i,\gamma,\upsilon$, the maps $(r,s,\omega)\mapsto\eta^{r,i,\gamma}_s(\omega)$, $(r,s,\omega)\mapsto\sigma^{r,i}_s(\omega)$, and $(r,s,\omega)\mapsto\tilde{A}^{r,i,\upsilon}_s(\omega)$ are measurable with respect to $(\mathcal{R}\otimes\mathcal{B}_{[0,T]})\otimes\mathcal{T}$, where $\mathcal{B}_{[0,T]}$ is the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathbf{R}\times[0,T]\times\Omega$ is identified elementwise with $(\mathbf{R}\times[0,T])\times\Omega$.

(b) (Event-time evaluations) For every $j\ge1$ and all $i,\gamma$: on the union of the cells of $\mathbf{R}$ with at least $j$ events (a member of $\mathcal{R}$), writing $t_j(r)$ for the $j$-th event time of $r$, the maps $(r,\omega)\mapsto\eta^{r,i,\gamma}_{t_j(r)-}(\omega)$ and $(r,\omega)\mapsto\tilde{A}^{r,i,\upsilon}_{t_j(r)}(\omega)$, the first defined via the left limit where it exists and $0$ elsewhere, are measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ (restricted to that union).

(c) (Regularity on the good set) For every $(r,\omega)\in G$: for every $i$ and $s$ there is exactly one $\gamma$ with $\eta^{r,i,\gamma}_s(\omega)=1$, the others being $0$, and $\sigma^{r,i}_s(\omega)$ is that $\gamma$; each path $s\mapsto\sigma^{r,i}_s(\omega)$ is piecewise constant and right-continuous in the sense of condition 1 of Solution of the Controlled N-Agent Dynamics, with $\sigma^{r,i}_0(\omega)=\varsigma^i_0(\omega)$, so in particular all left limits along these paths exist; and $\tilde{A}^{r,i,\upsilon}_s(\omega)=\int_{[0,s]}\tilde{\beta}\bigl(\sigma^{r,i}_u(\omega),\upsilon,\Sigma^r_u(\omega)\bigr)\,du,$ the Lebesgue integral over the compact interval $[0,s]$.

(d) (Per-record solution property) For every $r\in\mathbf{R}$ there is an event $\Omega^r\in\mathcal{F}$ with $P(\Omega^r)=1$ and $\{(r,\omega):\omega\in\Omega^r\}\subseteq G$ such that the state processes $\sigma^{r,i}$, the observation processes $\Upsilon^{r,\upsilon}_s=\frac{1}{N}\sum_{i=1}^N\tilde{Y}^{i,\upsilon}_{\tilde{A}^{r,i,\upsilon}_s}$, and the control process $s\mapsto a^r(s)$, together with the regular event $\Omega^r$, form a solution of the controlled $N$-agent dynamics on $[0,T]$ for the record-frozen policy $\hat{h}^r$ (which is $\mathcal{A}$-valued: by The Record-Frozen Control Path and Record-Frozen Policy every member of $\hat{h}^r$ takes the value $a^r(s)=h_{k_r(s)}\bigl(s,(t_1,\dots,t_{k_r(s)}),(v_1,\dots,v_{k_r(s)})\bigr)$, whose time argument lies in $R_{k_r(s)}(T)$, so this value lies in $\mathcal{A}$ because $h$ is $\mathcal{A}$-valued), whose consumed clock times for the observation clocks, denoted $\tilde{\mathcal{T}}^{i,\upsilon}$ in condition 2 of Solution of the Controlled N-Agent Dynamics, agree with the $\tilde{A}^{r,i,\upsilon}$ on $\Omega^r$.

(e) (Causality) Let $r,r'\in\mathbf{R}$ and $s\in[0,T]$ be such that $k_r(s)=k_{r'}(s)$ and the first $k_r(s)$ event times and marks of $r$ and $r'$ coincide. Then for every $\omega$ with $(r,\omega)\in G$ and $(r',\omega)\in G$: $\eta^{r,i,\gamma}_u(\omega)=\eta^{r',i,\gamma}_u(\omega)$ and $\tilde{A}^{r,i,\upsilon}_u(\omega)=\tilde{A}^{r',i,\upsilon}_u(\omega)$ for all $u\in[0,s]$ and all $i,\gamma,\upsilon$.

(f) (Record measurability and consistency) For every solution of the controlled $N$-agent dynamics on $[0,T]$ for the policy $h$ on this driving system, with occupation indicators $\eta^{i,\gamma}$, consumed observation clock times $\tilde{\mathcal{T}}^{i,\upsilon}$, observation filtration $(\mathcal{G}_s)_{s\in[0,T]}$, and observation record $W$: the map $W$ is measurable from $(\Omega,\mathcal{F})$ to $(\mathbf{R},\mathcal{R})$, with $W^{-1}(A)\in\mathcal{G}_T$ for every $A\in\mathcal{R}$; and there is an event of probability one on which: $(W(\omega),\omega)\in G$, and $\eta^{i,\gamma}_u(\omega)=\eta^{W(\omega),i,\gamma}_u(\omega)$ and $\tilde{\mathcal{T}}^{i,\upsilon}_u(\omega)=\tilde{A}^{W(\omega),i,\upsilon}_u(\omega)$ for all $u\in[0,T]$ and all $i,\gamma,\upsilon$.