Progressive Measurability of the Realized Control with Respect to the Observation Filtration

Adopt the setting, hypotheses and notation of the realized-control lemma: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$; a nonempty convex subset $\mathcal{A}$ of Euclidean space $\mathbb{R}^m$ which is compact for the topology determined by the Euclidean distance, with a real $R\ge0$ such that $|a|\le R$ for every $a\in\mathcal{A}$, where $|\cdot|$ is the Euclidean norm; a transition-rate family $\beta$ with control set $\mathcal{A}$; an observation-rate family $\tilde{\beta}$ with $\tilde{l}$ channels; a horizon $T>0$; an $N$-agent driving system $(\Omega,\mathcal{F},P)$; an $\mathcal{A}$-valued observation-driven control policy $h=(h_k)_{k\ge0}$ with record spaces $R_k(T)$ (a notation of the policy definition, unrelated to the bound $R$); a solution $(\sigma^i,\Upsilon^\upsilon,\alpha)$ on $[0,T]$ with regular event $\Omega_0$, observation counters $\tilde{N}^{i,\upsilon}$ indexed by agents $i\in\{1,\dots,N\}$ and channels $\upsilon\in\{1,\dots,\tilde{l}\}$, observation total $\tilde{c}=(\tilde{c}_t)_{t\in[0,T]}$, observation-event count $K_t=\tilde{c}_t$, observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, both filtrations with time index restricted to $[0,T]$ by clause (vii)(e) of the existence and uniqueness theorem; the event times $\tau_j$ and channels $\upsilon_j$ ($j\ge1$) of claim 1 of the realized-control lemma; the fixed point $a_0\in\mathcal{A}$; the realized control $\hat{\alpha}:[0,T]\times\Omega\to\mathcal{A}$ of claim 2 there; the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls with the metric $\rho$ of the weak metrizability and compactness theorem, formed from the sequence $(w_r)_{r\in\mathbb{N}}$ fixed in the realized-control lemma; and the convention of claim 3 there by which $\hat{\alpha}(\omega)$ denotes both the path $s\mapsto\hat{\alpha}(s,\omega)$ and the element of $\mathcal{U}_{\mathcal{A}}$ it represents. For $t\in[0,T]$ let $\mathcal{B}_{[0,t]}$ be the trace Borel $\sigma$-algebra on $[0,t]$ (for $t=0$ the $\sigma$-algebra $\{\emptyset,\{0\}\}$), let $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ be the product $\sigma$-algebra on $[0,t]\times\Omega$, and write $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,t]$, taken to be $0$ for $t=0$. A real-valued function on $\Omega$ is called $\mathcal{G}_t$-measurable when it is measurable with respect to $\mathcal{G}_t$ and the Borel $\sigma$-algebra of the real line; progressive measurability and stopping times refer to $(\mathcal{G}_t)_{t\in[0,T]}$ unless another filtration is named; and for a function $\theta:\Omega\to\mathbb{R}$ and a real $q$ we write $\{\theta\le q\}$ for $\{\omega\in\Omega:\theta(\omega)\le q\}$.

Then the following hold.

1. (Null events.) For every $t\in[0,T]$: $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$; every event of $\mathcal{F}$ of probability zero, and in particular $\Omega\setminus\Omega_0$, belongs to $\mathcal{G}_t$; and if $X$ is a random variable on $(\Omega,\mathcal{F},P)$ and $X'$ is a $\mathcal{G}_t$-measurable random variable with $X(\omega)=X'(\omega)$ for every $\omega$ outside some event of $\mathcal{F}$ of probability zero, then $X$ is $\mathcal{G}_t$-measurable.

2. (Event times and channels.) For every $j\ge1$ and every $q\in[0,T]$ one has $\{\tau_j\le q\}\in\mathcal{G}_q$. Consequently the function $\omega\mapsto\min(\tau_j(\omega),T)$ is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$. Moreover, for every $t\in[0,T]$, every $j\ge1$ and every channel $\upsilon\in\{1,\dots,\tilde{l}\}$,

$$\{\tau_j\le t\}\cap\{\upsilon_j=\upsilon\}\in\mathcal{G}_t .$$

3. (Progressive measurability of the realized control.) Every component $\hat{\alpha}^\kappa$ ($\kappa\in\{1,\dots,m\}$) of the realized control is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$, hence also with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$; in particular, for every $t\in[0,T]$, the restriction of $\hat{\alpha}^\kappa$ to $[0,t]\times\Omega$ is measurable with respect to $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ and the Borel $\sigma$-algebra of the real line, and $\hat{\alpha}^\kappa(t,\cdot)$ is $\mathcal{G}_t$-measurable.

4. (Truncated realized controls.) For $t\in[0,T]$ define $\hat{\alpha}^{(t)}:[0,T]\times\Omega\to\mathcal{A}$ by $\hat{\alpha}^{(t)}(s,\omega)=\hat{\alpha}(s,\omega)$ for $s\le t$ and $\hat{\alpha}^{(t)}(s,\omega)=a_0$ for $s>t$. Then for every $\omega\in\Omega$ the path $s\mapsto\hat{\alpha}^{(t)}(s,\omega)$ is an admissible representative, in the sense of claim 2 of the flow stability lemma, of an element of $\mathcal{U}_{\mathcal{A}}$, also written $\hat{\alpha}^{(t)}(\omega)$, and the paths $\hat{\alpha}^{(t)}(\omega)$ and $\hat{\alpha}(\omega)$ agree at every $s\in[0,t]$; for every $\zeta\in\mathcal{U}_{\mathcal{A}}$ the map $\omega\mapsto\rho\bigl(\hat{\alpha}^{(t)}(\omega),\zeta\bigr)$ is $\mathcal{G}_t$-measurable; and the map $\omega\mapsto\hat{\alpha}^{(t)}(\omega)$ is measurable with respect to $\mathcal{G}_t$ and the Borel $\sigma$-algebra of the metric space $(\mathcal{U}_{\mathcal{A}},\rho)$.

5. (The control energy process.) Let $A:[0,T]\to\mathcal{A}$ be a map each of whose components is measurable with respect to $\mathcal{B}_{[0,T]}$ and the Borel $\sigma$-algebra of the real line. Then the family $\bigl(|\hat{\alpha}(s,\cdot)-A_s|^2\bigr)_{s\in[0,T]}$ is progressively measurable and bounded by $4R^2$, and

$$\mathcal{E}_t(\omega)=\int_{[0,t]}\bigl|\hat{\alpha}(s,\omega)-A_s\bigr|^2\,ds\qquad(t\in[0,T],\ \omega\in\Omega)$$

defines a family $\mathcal{E}=(\mathcal{E}_t)_{t\in[0,T]}$ of $[0,4R^2T]$-valued functions on $\Omega$ which is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$, with $\mathcal{E}_t$ $\mathcal{G}_t$-measurable for every $t$, and such that $0\le\mathcal{E}_t(\omega)-\mathcal{E}_{t_0}(\omega)\le4R^2(t-t_0)$ for all $0\le t_0\le t\le T$ and every $\omega\in\Omega$; in particular every path $t\mapsto\mathcal{E}_t(\omega)$ is nondecreasing and continuous on $[0,T]$, the interval and the real line carrying the metric of the real line.