The Observation Record of a Solution of the Controlled N-Agent Dynamics

Adopt the setting of the controlled $N$-agent dynamics with control dimension $m\ge1$, and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$: a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$, an observation-rate family $\tilde{\beta}$ with $\tilde{l}$ observation channels, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, and a solution on $[0,T]$ with regular event $\Omega_0$, observation-event count $K_t$, observation event times $\tau_1<\dots<\tau_{K_T}$, and channels $\upsilon_1,\dots,\upsilon_{K_T}$, as in condition 5 of Solution of the Controlled N-Agent Dynamics, the channel $\upsilon_j$ being well defined at every $\omega\in\Omega_0$ by clause (vii)(d) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics.

For $\omega\in\Omega_0$ the value $W(\omega)$ below lies in $\mathbf{R}(T,\tilde{l})$: by condition 3 of Solution of the Controlled N-Agent Dynamics the observation total coincides on $[0,T]$ with the restriction of a counting path, so $K_T(\omega)$ is $0$ or a natural number, and the event times, which by condition 5 of Solution of the Controlled N-Agent Dynamics are the jump times of that path lying in $[0,T]$ listed in increasing order, satisfy $0<\tau_1(\omega)<\dots<\tau_{K_T(\omega)}(\omega)\le T$, the strict positivity because every jump time of a counting path is a real number $t>0$ by Counting Path and Its Jump Times; hence they form a point of the ordered time simplex $D_{K_T(\omega)}(T)$ when $K_T(\omega)\ge1$.

The observation record of the solution is the map $W:\Omega\to\mathbf{R}(T,\tilde{l})$ into the observation record space with horizon $T$ and $\tilde{l}$ channels defined by $W(\omega)=\bigl(K_T(\omega),\ (\tau_1(\omega),\dots,\tau_{K_T(\omega)}(\omega)),\ (\upsilon_1(\omega),\dots,\upsilon_{K_T(\omega)}(\omega))\bigr)$ for $\omega\in\Omega_0$ with $K_T(\omega)\ge1$, and by $W(\omega)=r_\emptyset$ otherwise, that is, for $\omega\in\Omega_0$ with $K_T(\omega)=0$ and for $\omega\notin\Omega_0$.