Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution

Adopt the setting and notation of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks: 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 $\mathbb{G}_N$ (a finite set with at most $(N+1)^l$ elements, since $Nx^\gamma\in\{0,\dots,N\}$ for $x\in\mathbb{G}_N$), the transition labels $c=(\sigma,\gamma)$ with their vectors $v_c$, clock families $p=(p^{c})$ of counting paths, control paths $a$, and open-loop aggregate solutions with their consumed clock times $\mathsf{C}^{c}$ and transition counters $\mathsf{N}^{c}$; coordinates of points of $\mathbb{R}^l$ are written with superscripts, $x=(x^1,\dots,x^l)$. Write $\mathbb{N}_0$ for the set consisting of $0$ and the natural numbers, $\tau_j(q)$ for the $j$-th jump time of a counting path $q$ (with $\tau_j(q)=+\infty$ allowed, as in Counting Path and Its Jump Times), $\mathcal{B}_{[0,t]}$ for the trace Borel $\sigma$-algebra on $[0,t]$, $\lambda_{[0,T]}$ for the restricted Lebesgue measure on $[0,T]$, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\otimes$ for the product $\sigma$-algebra of two $\sigma$-algebras and for the product measure of two $\sigma$-finite measures (products of three factors being associated to the right), $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,t]$ (equal to $0$ for $t=0$), and $\mathbf{1}_{E}$ for the function equal to $1$ on a set $E$ and $0$ off $E$. Greatest lower bounds and least upper bounds of sets of real numbers are those of Lower Bound and Greatest Lower Bound and Upper Bound and Least Upper Bound (existing for nonempty sets bounded below, respectively above, by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below and the completeness of the real numbers); the greatest lower bound of the empty set is $+\infty$; a $\lambda_{[0,T]}$-null set is a member $Z$ of $\mathcal{B}_{[0,T]}$ with $\lambda_{[0,T]}(Z)=0$; and minima of finitely many elements of $[0,+\infty]$ are taken in $[0,+\infty]$, where $+\infty$ exceeds every real number. Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

Fix a clock family $p$, a control path $a$ and a point $x_0\in\mathbb{G}_N$.

The aggregate recursion. Put $\theta_0=0$, $x^{(0)}=x_0$ and $\kappa^{c}_0=0$ for every label $c$. Given $k\in\mathbb{N}_0$ together with $\theta_k\in[0,T)$, $x^{(k)}\in\mathbb{G}_N$ and real numbers $\kappa^{c}_k\ge0$ (one for each label), define for every label $c=(\sigma,\gamma)$ and every $t\in[0,T]$

let $\lambda^{c}_k=\tau_{j}(p^{c})$ with $j=p^{c}(\kappa^{c}_k)+1$ (the first jump time of $p^{c}$ above the level $\kappa^{c}_k$), let $h^{c}_k$ be the greatest lower bound of $\{t\in[\theta_k,T]:\ \mathsf{C}^{c,(k)}_t\ge\lambda^{c}_k\}$, let

let $\mathcal{J}_k$ be the set of labels $c$ with $\mathsf{C}^{c,(k)}_{\theta_{k+1}}\ge\lambda^{c}_k$, and put

The recursion stops at the first index $K$ for which $\theta_K=T$ or $x^{(K)}\notin\mathbb{G}_N$; the data $(p,a,x_0)$ are called conflict-free if it stops with $\theta_K=T$ and $x^{(K)}\in\mathbb{G}_N$. The recursion path is the map $\Sigma^{\mathrm{rec}}:[0,T]\to\mathbb{R}^l$ with $\Sigma^{\mathrm{rec}}_t=x^{(k)}$ for $t\in[\theta_k,\theta_{k+1})$ and $k<K$, and $\Sigma^{\mathrm{rec}}_t=x^{(K)}$ for $t\in[\theta_K,T]$; the recursion consumed times and counters are $\mathsf{C}^{\mathrm{rec},c}_t=\mathsf{C}^{c,(k)}_t$ for $t\in[\theta_k,\theta_{k+1}]$ and $k<K$, $\mathsf{C}^{\mathrm{rec},c}_t=\kappa^{c}_K$ for $t\in[\theta_K,T]$, and $\mathsf{N}^{\mathrm{rec},c}_t=p^{c}(\mathsf{C}^{\mathrm{rec},c}_t)$ (the two prescriptions for $\mathsf{C}^{\mathrm{rec},c}_t$ at $t=\theta_{k+1}$ agree, since $\mathsf{C}^{c,(k+1)}_{\theta_{k+1}}=\kappa^{c}_{k+1}$ by claim 1 applied at step $k+1$ whenever that step is performed).

1. (The recursion is well defined and stops.) For every $k$ at which the recursion has not yet stopped, each integrand above is measurable on $[0,T]$ with respect to $\mathcal{B}_{[0,T]}$ and $\mathcal{B}(\mathbb{R})$ and takes values in $[0,NB]$; each map $t\mapsto\mathsf{C}^{c,(k)}_t$ is nondecreasing, satisfies $\mathsf{C}^{c,(k)}_{\theta_k}=\kappa^{c}_k$ and $|\mathsf{C}^{c,(k)}_t-\mathsf{C}^{c,(k)}_s|\le NB(t-s)$ for $0\le s\le t\le T$, hence is continuous on $[0,T]$; $\lambda^{c}_k>\kappa^{c}_k$; $\theta_{k+1}>\theta_k$; if $\theta_{k+1}<T$ then $\mathcal{J}_k$ is nonempty; $\mathsf{C}^{c,(k)}_{\theta_{k+1}}=\lambda^{c}_k$ for every $c\in\mathcal{J}_k$; and $\kappa^{c}_{k+1}\le NB\,\theta_{k+1}$ and $\mathsf{C}^{c,(k)}_t\le NBt$ for $t\in[\theta_k,T]$, for every label $c$. The recursion stops at some index $K\le1+\sum_{c}p^{c}(NBT)$, the sum running over all labels.

2. (Uniqueness, and existence for conflict-free data.) Any two open-loop aggregate solutions on $[0,T]$ for the data $(p,a,x_0)$ coincide. An open-loop aggregate solution exists if and only if the data are conflict-free, and in that case the unique solution $\Sigma$ equals the recursion path $\Sigma^{\mathrm{rec}}$, its consumed clock times and counters are the recursion consumed times and counters, and, for every label $c$:

(a) $0\le\mathsf{C}^{c}_s\le\mathsf{C}^{c}_t\le NBt$ and $|\mathsf{C}^{c}_t-\mathsf{C}^{c}_s|\le NB(t-s)$ for all $0\le s\le t\le T$, so that $t\mapsto\mathsf{C}^{c}_t$ is continuous on $[0,T]$;

(b) $t\mapsto\mathsf{N}^{c}_t$ is nondecreasing on $[0,T]$ with values in $\mathbb{N}_0$, $\mathsf{N}^{c}_0=0$, $\mathsf{N}^{c}_t\le p^{c}(NBt)$ for every $t$, and $\mathsf{N}^{c}_t$ is the greatest lower bound of $\{\mathsf{N}^{c}_s:\ t<s\le T\}$ for every $t\in[0,T)$;

(c) $\Sigma$ is constant on each interval $[\theta_k,\theta_{k+1})$ with $k<K$; for every $k<K$ one has $\Sigma_{\theta_{k+1}}-\Sigma_s=x^{(k+1)}-x^{(k)}=\frac{1}{N}\sum_{c\in\mathcal{J}_k}v_c$ for all $s\in[\theta_k,\theta_{k+1})$ (a difference which may vanish, so that not every $\theta_k$ need be a discontinuity of $\Sigma$), and $\mathsf{C}^{c}_{\theta_{k+1}}$ is a jump time of $p^{c}$ for every $c\in\mathcal{J}_k$; consequently every $t\in(0,T]$ with $\Sigma_t\neq\Sigma_s$ for all $s\in[0,t)$ sufficiently close to $t$ is one of $\theta_1,\dots,\theta_K$, and there are at most $\sum_{c}p^{c}(NBT)$ such times.

3. (Causality.) Let $t\in[0,T]$, let $p'$ be a clock family with $p'^{c}(u)=p^{c}(u)$ for every label $c$ and every $u\in[0,NBt]$, and let $a'$ be a control path such that $\{s\in[0,t]:a'_s\neq a_s\}$ is contained in a $\lambda_{[0,T]}$-null set. Run the recursions for $(p,a,x_0)$ and $(p',a',x_0)$, marking the quantities of the second by a prime. Then for every $k$ such that neither recursion has stopped before step $k$ and $\min(\theta_k,\theta'_k)\le t$, one has $\theta_k=\theta'_k$, $x^{(k)}=x'^{(k)}$ and $\kappa^{c}_k=\kappa'^{c}_k$ for every $c$; in particular, if one of the recursions stops at an index $K$ with $\theta_K\le t$, then so does the other, with the same $x^{(K)}$ and $\kappa^{c}_K$. Consequently $\Sigma^{\mathrm{rec}}_s=\Sigma'^{\mathrm{rec}}_s$, $\mathsf{C}^{\mathrm{rec},c}_s=\mathsf{C}'^{\mathrm{rec},c}_s$ and $\mathsf{N}^{\mathrm{rec},c}_s=\mathsf{N}'^{\mathrm{rec},c}_s$ for every $s\in[0,t]$ and every label; and if both data are conflict-free, the solutions satisfy $\Sigma_s=\Sigma'_s$, $\mathsf{C}^{c}_s=\mathsf{C}'^{c}_s$ and $\mathsf{N}^{c}_s=\mathsf{N}'^{c}_s$ for every $s\in[0,t]$ and every label.

4. (Measurability in the clocks and in a parameter.) 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 (for instance a family of aggregate transition clocks; neither the Poisson law nor independence is used in this claim), let $(\mathsf{R},\mathcal{R})$ be a nonempty measurable space, and let $\mathsf{a}:[0,T]\times\mathsf{R}\to\mathcal{A}$ be a map each of whose components is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ and $\mathcal{B}(\mathbb{R})$. Then for every $r\in\mathsf{R}$ the map $s\mapsto\mathsf{a}(s,r)$ is a control path. For $r\in\mathsf{R}$ and $\omega\in\Omega$ run the recursion for the data $(\mathsf{P}(\omega),\mathsf{a}(\cdot,r),x_0)$, where $\mathsf{P}(\omega)$ denotes the clock family $(u\mapsto\mathsf{P}^{c}_u(\omega))$; write $\Sigma^{r}_t(\omega)$, $\mathsf{C}^{r,c}_t(\omega)$ and $\mathsf{N}^{r,c}_t(\omega)$ for its recursion path, recursion consumed times and recursion counters, and $\mathsf{G}\subseteq\mathsf{R}\times\Omega$ for the set of pairs $(r,\omega)$ whose data are conflict-free. For $t\in[0,T]$ let $\mathcal{H}_t$ be the $\sigma$-algebra generated by the random variables $\mathsf{P}^{c}_u$ with $u\in[0,NBt]$ and $c$ ranging over all labels. Then $\mathsf{G}\in\mathcal{R}\otimes\mathcal{H}_T$, and for every $t\in[0,T]$, every $\gamma\in\{1,\dots,l\}$ and every label $c$, each of the maps

is measurable with respect to $\mathcal{R}\otimes\mathcal{H}_t$ and $\mathcal{B}(\mathbb{R})$, hence with respect to $\mathcal{R}\otimes\mathcal{F}$; and each of the maps $(t,r,\omega)\mapsto\Sigma^{r,\gamma}_t(\omega)$, $(t,r,\omega)\mapsto\mathsf{C}^{r,c}_t(\omega)$ and $(t,r,\omega)\mapsto\mathsf{N}^{r,c}_t(\omega)$ on $[0,T]\times\mathsf{R}\times\Omega$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$ and $\mathcal{B}(\mathbb{R})$. In particular, when $\mathsf{R}$ is a one-point set, so that $\mathsf{a}$ is a single control path $a$, and the data $(\mathsf{P}(\omega),a,x_0)$ are conflict-free for every $\omega\in\Omega$, the family $(\Sigma_t)_{t\in[0,T]}$ with $\Sigma_t(\omega)=\Sigma^{r}_t(\omega)$, $r$ being the unique point of $\mathsf{R}$, is the only open-loop aggregate solution driven by $\mathsf{P}$ for $a$ and $x_0$ (when $\mathsf{P}$ is a family of aggregate transition clocks), and each $\Sigma^\gamma_t$ is an $\mathcal{H}_t$-measurable random variable.