Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon

Adopt the setting and notation of Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels (that of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution): natural numbers $N\ge1$, $l\ge2$, $m\ge1$, a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ in Euclidean space, real numbers $B\ge0$ and $T>0$, a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$, the aggregate lattice $\mathbb{G}_N$ (a nonempty finite set), the transition labels $c=(\sigma,\gamma)$ with vectors $v_c$, a fixed control path $a$ and a fixed point $x_0\in\mathbb{G}_N$. Let $R$ be a real number or $+\infty$ with $R>NBT$, and let $(\Omega,\mathcal{F},P)$ be a probability space carrying a family $\mathsf{P}=(\mathsf{P}^{c})$, indexed by the transition labels, such that each $\mathsf{P}^{c}$ is a Poisson clock with horizon $R$ and the family of $\sigma$-algebras $\sigma(\mathsf{P}^{c}_u:u\ge0)$, indexed by the labels, is independent (for $R=+\infty$ this says that $\mathsf{P}$ is a family of aggregate transition clocks; for finite $R>NBT$ the copy clocks of The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them qualify, that lemma allowing any $R\ge NBT$). For $\omega\in\Omega$ run the aggregate recursion for the data $(\mathsf{P}(\omega),a,x_0)$, with recursion path $\Sigma_t(\omega)$, recursion consumed times $\mathsf{C}^{c}_t(\omega)$, recursion counters $\mathsf{N}^{c}_t(\omega)$ and recursion filtration $(\mathfrak{F}_t)_{t\in[0,T]}$ as in Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels, and let $\Omega_0$ be the event that the data $(\mathsf{P}(\omega),a,x_0)$ are conflict-free (an event by claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, applied with a one-point parameter space). Assume $P(\Omega_0)=1$.

For a label $c=(\sigma,\gamma)$, $u\in[0,T]$ and $x\in\mathbb{G}_N$ put

$$g^{c}(u,x)=N\,x^{\sigma}\,\beta(\sigma,\gamma,x,a_u)\in[0,NB],\qquad \phi^{c}(x)=\begin{cases}x+\tfrac{1}{N}v_c&\text{if }x+\tfrac{1}{N}v_c\in\mathbb{G}_N,\\ x&\text{otherwise,}\end{cases}$$

and for $u\in[0,T]$ and $x\neq y$ in $\mathbb{G}_N$ put $q_u(x,y)=\sum_{c:\,\phi^{c}(x)=y}g^{c}(u,x)$, which equals $Nx^{\sigma}\beta(\sigma,\gamma,x,a_u)$ if $y=x+\frac1Nv_c$ for the (then unique) label $c=(\sigma,\gamma)$, and $0$ if $y-x$ is not of the form $\frac1Nv_c$. Finally let $\bar{\Sigma}_t(\omega)=\Sigma_t(\omega)$ if $\Sigma_t(\omega)\in\mathbb{G}_N$ and $\bar{\Sigma}_t(\omega)=x_0$ otherwise (so $\bar{\Sigma}=\Sigma$ on $\Omega_0$, where the recursion path stays in $\mathbb{G}_N$). Then:

1. (The recursion is a Poisson-clock jump system.) The data consisting of $(\Omega,\mathcal{F},P)$, $T$, $\Lambda=NB$, the state space $E=\mathbb{G}_N$, the clock labels $\mathsf{A}=$ the transition labels, the clocks $Y^{c}=\mathsf{P}^{c}$, the rate functions $g^{c}$, the transition maps $\phi^{c}$, the filtration $(\mathfrak{F}_t)$, the event $\Omega_0$, the state process $X_t=\bar{\Sigma}_t$ and the consumed clock times $\mathcal{T}^{c}_t=\mathsf{C}^{c}_t$ satisfy (D1)--(D4), (H1) and (H2) of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property; the counters of that lemma are the recursion counters $\mathsf{N}^{c}_t$.

2. (Forward equation.) Consequently, for every $F:\mathbb{G}_N\to\mathbb{R}$, all $0\le r\le t\le T$ and every $D\in\mathfrak{F}_r$,

$$\mathbb{E}\bigl[(F(\bar{\Sigma}_t)-F(\bar{\Sigma}_r))\mathbf{1}_D\bigr]=\mathbb{E}\Bigl[\mathbf{1}_D\int_{[r,t]}\mathbf{1}_{\Omega_0}\sum_{c=(\sigma,\gamma)}N\bar{\Sigma}^{\sigma}_u\beta(\sigma,\gamma,\bar{\Sigma}_u,a_u)\bigl(F(\bar{\Sigma}_u+\tfrac1Nv_c)-F(\bar{\Sigma}_u)\bigr)\,du\Bigr],$$

where $F(\bar{\Sigma}_u+\frac1Nv_c)$ is read as $F(\bar{\Sigma}_u)$ when $\bar{\Sigma}_u+\frac1Nv_c\notin\mathbb{G}_N$ (in which case $\bar{\Sigma}^{\sigma}_u=0$, so the term vanishes anyway); and for every $r\in[0,T)$ and $D\in\mathfrak{F}_r$ the family $\mu^{D}_u(x)=P(D\cap\{\bar{\Sigma}_u=x\})$ ($u\in[r,T]$, $x\in\mathbb{G}_N$) is a solution of the forward equation on $[r,T]$ for the rates $q$ with rate bound $l(l-1)NB$.