Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels

Adopt the setting and notation of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution: natural numbers $N\ge1$, $l\ge2$, $m\ge1$, a nonempty subset $\mathcal{A}$ of Euclidean space $\mathbb{R}^m$, real numbers $B\ge0$ and $T>0$, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, the aggregate lattice $\mathbb{G}_N$, the transition labels $c$, clock families $p=(p^{c})$ of counting paths and control paths $a$, and, for data $(p,a,x_0)$ with $x_0\in\mathbb{G}_N$, the aggregate recursion of that theorem with its steps $\theta_k$, $x^{(k)}$, $\kappa^{c}_k$, $\mathsf{C}^{c,(k)}$, $\lambda^{c}_k$, $h^{c}_k$, $\mathcal{J}_k$, its stopping index $K$, its recursion path $\Sigma^{\mathrm{rec}}$, and its recursion consumed times $\mathsf{C}^{\mathrm{rec},c}$ and counters $\mathsf{N}^{\mathrm{rec},c}$. A family $\mathbf{w}=(w_c)_c$ of nonnegative real numbers indexed by the transition labels is called a family of caps. Fix a control path $a$ and a point $x_0\in\mathbb{G}_N$.

1. (Pathwise clock-reading bound.) Let $p$ and $p'$ be clock families and $\mathbf{w}$ a family of caps such that $p^{c}(u)=p'^{c}(u)$ for every label $c$ and every $u\in[0,w_c]$, and let $r\in[0,T]$ be such that the recursion for $(p,a,x_0)$ satisfies $\mathsf{C}^{\mathrm{rec},c}_r\le w_c$ for every label $c$. Mark the quantities of the recursion for $(p',a,x_0)$ by a prime. Then for every $k\le K$ with $\theta_k\le r$ one has $k\le K'$, $\theta_k=\theta'_k$, $x^{(k)}=x'^{(k)}$ and $\kappa^{c}_k=\kappa'^{c}_k$ for every $c$; if $\theta_K\le r$ then $K'=K$; and

$$\Sigma^{\mathrm{rec}}_s=\Sigma'^{\mathrm{rec}}_s,\qquad \mathsf{C}^{\mathrm{rec},c}_s=\mathsf{C}'^{\mathrm{rec},c}_s,\qquad \mathsf{N}^{\mathrm{rec},c}_s=\mathsf{N}'^{\mathrm{rec},c}_s\qquad\text{for all }s\in[0,r]\text{ and all labels }c .$$

In particular $\mathsf{C}'^{\mathrm{rec},c}_r\le w_c$ for every $c$, so the hypothesis on $r$ is symmetric in $p$ and $p'$.

2. (Clock-reading bound for the recursion driven by random clocks.) Let $(\Omega,\mathcal{F},P)$ be a probability space carrying a family $\mathsf{P}=(\mathsf{P}^{c})$, indexed by the transition labels, of stochastic processes $\mathsf{P}^{c}=(\mathsf{P}^{c}_u)_{u\ge0}$ all of whose paths are counting paths (no law is prescribed). For $\omega\in\Omega$ run the recursion for the data $(\mathsf{P}(\omega),a,x_0)$ and write $\Sigma_t(\omega)$, $\mathsf{C}^{c}_t(\omega)$ and $\mathsf{N}^{c}_t(\omega)$ for its recursion path, recursion consumed times and recursion counters; by claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, applied with a one-point parameter space (its measurability requirement on the control then being the measurability of $a$, and the conclusion being measurability in $\omega$ after taking sections), each $\Sigma^{\gamma}_t$, $\mathsf{C}^{c}_t$ and $\mathsf{N}^{c}_t$ is a random variable; conflict-freeness is not assumed. For $t\in[0,T]$ let $\mathfrak{F}_t$ be the $\sigma$-algebra generated by the random variables $\Sigma^{\gamma}_s$, $\mathsf{C}^{c}_s$ and $\mathsf{N}^{c}_s$ with $s\in[0,t]$, $\gamma\in\{1,\dots,l\}$ and $c$ ranging over the labels (the recursion filtration, a filtration; $\mathfrak{F}_s\subseteq\mathfrak{F}_t$ for $s\le t$). For a family of caps $\mathbf{w}$ let $\mathcal{K}_{\mathbf{w}}$ be the $\sigma$-algebra generated by the random variables $\mathsf{P}^{c}_u$ with $0\le u\le w_c$ and $c$ ranging over the labels (a different object from the $\sigma$-algebras $\mathcal{H}_t$ of claim 4 of the adopted theorem). Then for every $r\in[0,T]$ and every family of caps $\mathbf{w}$, with $C_{\mathbf{w}}$ the event that $\mathsf{C}^{c}_r\le w_c$ for every label $c$:

(a) $C_{\mathbf{w}}\in\mathcal{K}_{\mathbf{w}}$;

(b) for every $F\in\mathfrak{F}_r$ there is an $H\in\mathcal{K}_{\mathbf{w}}$ with $F\cap C_{\mathbf{w}}=H\cap C_{\mathbf{w}}$.

Consequently, for every $r\in[0,T]$, the consumed times $\mathsf{C}^{c}_r$ (which are $\mathfrak{F}_r$-measurable and satisfy $0\le\mathsf{C}^{c}_r\le NBr$) satisfy the clock-reading bound of Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound with the past $\mathfrak{F}=\mathfrak{F}_r$, the trivial initial data $\mathcal{I}=\{\emptyset,\Omega\}$, the clocks $Y^{c}=\mathsf{P}^{c}$ (indexed by the nonempty finite set of transition labels) and the consumed levels $\mathcal{T}^{c}=\mathsf{C}^{c}_r$. Only the clock-reading bound is asserted here: the remaining hypotheses of that lemma (that the $\mathsf{P}^{c}$ are Poisson clocks with a horizon $R$, that they are independent, and that $NBr<R$) concern the law of $\mathsf{P}$ and must be supplied separately.