Fresh-Start Property of the Controlled N-Agent Dynamics

Adopt the setting of the controlled $N$-agent dynamics: a transition-rate family $\beta$ with rate bound $B$, an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$, and a solution of the controlled $N$-agent dynamics on $[0,T]$, which exists by the existence and uniqueness theorem, with consumed clock times $A^{i,\sigma\gamma}_t$, $\tilde{A}^{i,\upsilon}_t$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$.

Fix $r\in[0,T]$ and define the residual clocks

for all $i\in\{1,\dots,N\}$, ordered pairs $(\sigma,\gamma)$ with $\sigma\neq\gamma$, and $\upsilon\in\{1,\dots,\tilde{l}\}$. Then:

(a) Every path of every residual clock is a counting path, and every residual clock is a homogeneous Poisson process with rate $1$ on $(\Omega,\mathcal{F},P)$.

(b) The family of $\sigma$-algebras consisting of $\mathcal{F}^{\mathrm{sys}}_r$ together with the $\sigma$-algebras $\sigma(\hat{Y}^{i,\sigma\gamma}_u:u\ge0)$, one for each transition clock, and $\sigma(\hat{\tilde{Y}}^{i,\upsilon}_u:u\ge0)$, one for each observation clock, is independent.

In particular, $(\Omega,\mathcal{F},P)$ equipped with the initial states $\sigma^1_r,\dots,\sigma^N_r$ and the residual clocks is again an $N$-agent driving system, since the time-$r$ states are $\mathcal{F}^{\mathrm{sys}}_r$-measurable by the adaptedness assertion of the existence and uniqueness theorem.