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

Adopt the setting of the controlled $N$-agent dynamics: a transition-rate family $\beta$, 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 $h$, 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.

For $\omega\in\Omega_0$ the value $W(\omega)$ below is an observation record: the count $K_T(\omega)$ is either $0$ or a natural number, since the observation total coincides on $[0,T]$ with the restriction of a counting path, whose values are $0$ or natural numbers; the event times $\tau_1(\omega)<\dots<\tau_{K_T(\omega)}(\omega)$ are strictly increasing members of $[0,T]$ by condition 5 of Solution of the Controlled N-Agent Dynamics; and $\tau_1(\omega)>0$ because, by Counting Path and Its Jump Times, every jump time of a counting path is a real number $t>0$. Hence the tuple of event times lies in 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 $\omega\notin\Omega_0$.