The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood

Adopt the setting and notation of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks and 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 $\mathbb{G}_N$ (a finite subset of the probability simplex $\Delta^l$), the transition labels $c$, clock families of counting paths, control paths, and, for data $(p,a,x_0)$, the recursion with its stopping index $K$, times $0=\theta_0<\theta_1<\dots<\theta_K\le T$, points $x^{(0)},\dots,x^{(K)}$, recursion path $\Sigma^{\mathrm{rec}}$, and the notion of conflict-free data. Let $\tilde{l}\ge1$ be a natural number, let $\tilde\beta$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$ (a real number with $\tilde{B}\ge0$), let $\tilde{b}=(\tilde{b}^\upsilon)_{\upsilon=1}^{\tilde{l}}$ be its aggregate observation drift, and put $\tilde{b}^{\mathrm{tot}}=\sum_{\upsilon=1}^{\tilde{l}}\tilde{b}^\upsilon$ on $\Delta^l$. Let $(\mathbf{R},\mathcal{R},\rho)=(\mathbf{R}(T,\tilde{l}),\mathcal{R}(T,\tilde{l}),\rho(T,\tilde{l}))$ be the observation record space with horizon $T$ and $\tilde{l}$ channels, records written $r=(k,\mathbf{t},v)$ with $\mathbf{t}=(t_1,\dots,t_k)$ and $v=(v_1,\dots,v_k)$, with cells $C_\emptyset$ and $C_{k,v}$ and with $\mathbf{R}^{(k)}$ the set of records with exactly $k$ events; let $\pi_{s-}$ ($s\in[0,T]$) be the strict prefix maps. Let $h=(h_k)_{k\ge0}$ be an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, all of whose members take values in $\mathcal{A}$, and for $r\in\mathbf{R}$ let $a^r$ be its record-frozen control path at $r$, with event count $k_r$. 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, write $\mathsf{P}(\omega)$ for the clock family $(u\mapsto\mathsf{P}^{c}_u(\omega))$, and let $\mathcal{H}_t$ ($t\in[0,T]$) be the $\sigma$-algebra generated by the variables $\mathsf{P}^{c}_u$ with $u\in[0,NBt]$ and $c$ ranging over all labels, as in claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution. Fix $x_0\in\mathbb{G}_N$.

Write $\mathcal{B}_{[0,T]}$ for the trace Borel $\sigma$-algebra on $[0,T]$, $\int_{[0,T]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,T]$, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\otimes$ for the product $\sigma$-algebra, and $\exp$ for the real exponential function; measurability of real-valued maps is always with respect to $\mathcal{B}(\mathbb{R})$ on the target. Coordinates of points of $\mathbb{R}^l$ carry superscripts, $x=(x^1,\dots,x^l)$.

1. (Joint measurability of the record-frozen control) Each component of the map $\mathsf{a}:[0,T]\times\mathbf{R}\to\mathcal{A}$, $\mathsf{a}(s,r)=a^r(s)$, is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$. Consequently claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution applies with $(\mathsf{R},\mathcal{R})=(\mathbf{R},\mathcal{R})$; for $r\in\mathbf{R}$ and $\omega\in\Omega$ write $\Sigma^{r}_t(\omega)$ for the recursion path of the data $(\mathsf{P}(\omega),a^r,x_0)$, and $\mathsf{G}\in\mathcal{R}\otimes\mathcal{H}_T$ for the set of pairs $(r,\omega)$ whose data are conflict-free.

2. (Regularised recursion path and its left limits) For $r\in\mathbf{R}$, $\omega\in\Omega$ and $t\in[0,T]$ put $\bar\Sigma^{r}_t(\omega)=\Sigma^{r}_t(\omega)$ if $\Sigma^{r}_t(\omega)\in\mathbb{G}_N$ and $\bar\Sigma^{r}_t(\omega)=x_0$ otherwise. Then, for every $(r,\omega)$, with $K$ and $\theta_0,\dots,\theta_K$ those of the recursion for $(\mathsf{P}(\omega),a^r,x_0)$:

(a) $\bar\Sigma^{r}_t(\omega)\in\mathbb{G}_N$ for every $t$; the map $t\mapsto\bar\Sigma^{r}_t(\omega)$ is constant on $[\theta_k,\theta_{k+1})$ for each $k<K$ and constant on $[\theta_K,T]$, and it coincides with $\Sigma^{r}(\omega)$ on $[0,\theta_K)$;

(b) for every $t\in(0,T]$ there is exactly one point $\bar\Sigma^{r}_{t-}(\omega)\in\mathbb{G}_N$, the left limit, for which there is a real number $\delta>0$ such that $\bar\Sigma^{r}_s(\omega)=\bar\Sigma^{r}_{t-}(\omega)$ for all $s\in[t-\delta,t)\cap[0,T]$; put $\bar\Sigma^{r}_{0-}(\omega)=x_0$; the set of $t\in[0,T]$ with $\bar\Sigma^{r}_{t-}(\omega)\neq\bar\Sigma^{r}_t(\omega)$ is contained in $\{\theta_1,\dots,\theta_K\}$;

(c) if $(r,\omega)\in\mathsf{G}$, then $\bar\Sigma^{r}_t(\omega)=\Sigma^{r}_t(\omega)$ for every $t$, and $t\mapsto\Sigma^{r}_t(\omega)$ is the unique open-loop aggregate solution on $[0,T]$ for the data $(\mathsf{P}(\omega),a^r,x_0)$;

(d) for every $\gamma\in\{1,\dots,l\}$ the maps $(t,r,\omega)\mapsto\bar\Sigma^{r,\gamma}_t(\omega)$ and $(t,r,\omega)\mapsto\bar\Sigma^{r,\gamma}_{t-}(\omega)$ on $[0,T]\times\mathbf{R}\times\Omega$ are measurable with respect to $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$, and for every fixed $\omega\in\Omega$ the maps $(t,r)\mapsto\bar\Sigma^{r,\gamma}_t(\omega)$ and $(t,r)\mapsto\bar\Sigma^{r,\gamma}_{t-}(\omega)$ on $[0,T]\times\mathbf{R}$ are measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$.

3. (The record-driven intensity is causal) For $\omega\in\Omega$, $\upsilon\in\{1,\dots,\tilde{l}\}$, $t\in[0,T]$ and $r\in\mathbf{R}$ put $\lambda^{\omega,\upsilon}_t(r)=N\,\tilde{b}^\upsilon\bigl(\bar\Sigma^{r}_{t-}(\omega)\bigr).$ Then for every $\omega$ the family $\lambda^{\omega}=(\lambda^{\omega,\upsilon})_{\upsilon}$ is a causal intensity on $\mathbf{R}$ with bound $N\tilde{B}$, whose total intensity is $\lambda^{\omega,\mathrm{tot}}_t(r)=N\tilde{b}^{\mathrm{tot}}(\bar\Sigma^{r}_{t-}(\omega))$. The non-anticipation rests on the following identity: for every $t\in[0,T]$, every $r\in\mathbf{R}$ and every $\omega$, $\bar\Sigma^{\pi_{t-}(r)}_u(\omega)=\bar\Sigma^{r}_u(\omega)\qquad\text{for all }u\in[0,t),\qquad\text{hence}\qquad \bar\Sigma^{\pi_{t-}(r)}_{t-}(\omega)=\bar\Sigma^{r}_{t-}(\omega).$ If moreover $\underline{b}\ge0$ is a real number with $\tilde{b}^\upsilon(x)\ge\underline{b}$ for all $x\in\Delta^l$ and all $\upsilon$, then $\lambda^{\omega,\upsilon}_t(r)\ge N\underline{b}$ for all $\omega,\upsilon,t,r$.

4. (The record-driven likelihood) Let $\ell^{\omega}=\ell_{\lambda^{\omega}}$ be the likelihood of $\lambda^{\omega}$. Then the map $(r,\omega)\mapsto\ell^{\omega}(r)$ on $\mathbf{R}\times\Omega$ is measurable with respect to $\mathcal{R}\otimes\mathcal{F}$; $0\le\ell^{\omega}(r)\le(N\tilde{B})^{k}$ for $r\in\mathbf{R}^{(k)}$ (with $(N\tilde{B})^0=1$); $\int_{\mathbf{R}}\ell^{\omega}\,d\rho=1$ for every $\omega\in\Omega$; and for every $(r,\omega)\in\mathsf{G}$ with $r=(k,\mathbf{t},v)$, $\ell^{\omega}(r)=\Bigl(\prod_{i=1}^{k}N\,\tilde{b}^{v_i}\bigl(\Sigma^{r}_{t_i-}(\omega)\bigr)\Bigr)\exp\Bigl(-N\int_{[0,T]}\tilde{b}^{\mathrm{tot}}\bigl(\Sigma^{r}_s(\omega)\bigr)\,ds\Bigr),$ where $\Sigma^{r}(\omega)$ is the open-loop aggregate solution of claim 2(c), $\Sigma^{r}_{t_i-}(\omega)=\bar\Sigma^{r}_{t_i-}(\omega)$ its left limit, and the finite product is $1$ when $k=0$.