Proof of The Open-Loop Policy Determined by a Measurable Control

Claim 1. By the definition of an observation-driven control policy we must check that each component of $h^A_0$ is measurable on $[0,T]$, and that for each $k\ge1$, each channel tuple $(\upsilon_1,\dots,\upsilon_k)$ and each component index $j$, the map $(t,\tau_1,\dots,\tau_k)\mapsto h^{A,j}_k(t,\tau,\upsilon)$ is measurable on $[0,T]\times R_k(T)$ for the $\sigma$-algebra generated by the relatively open subsets, where $R_k(T)=\{(\tau_1,\dots,\tau_k):0\le\tau_1\le\dots\le\tau_k\le T\}$.

The first requirement is the hypothesis on $A$. For the second, fix $k\ge1$, $j$ and a channel tuple, and let $\mathcal{E}$ be the collection of subsets $E\subseteq[0,T]$ such that $E\times R_k(T)$ belongs to the $\sigma$-algebra generated by the relatively open subsets of $[0,T]\times R_k(T)$. Then $\mathcal{E}$ is a $\sigma$-algebra on $[0,T]$: it contains $[0,T]$; it is closed under complements, since $([0,T]\setminus E)\times R_k(T)$ is the complement of $E\times R_k(T)$ in $[0,T]\times R_k(T)$; and it is closed under countable unions, since $\big(\bigcup_iE_i\big)\times R_k(T)=\bigcup_i\big(E_i\times R_k(T)\big)$. Moreover $\mathcal{E}$ contains every relatively open $E\subseteq[0,T]$, because then $E\times R_k(T)$ is relatively open in the product. Hence $\mathcal{E}$ contains the $\sigma$-algebra generated by the relatively open subsets of $[0,T]$, that is the trace Borel $\sigma$-algebra.

Now for every real number $c$,

and $\{t:A^j_t>c\}$ is a trace Borel set by the hypothesis on $A$ together with the generator criterion of measurability. By the previous paragraph the product lies in the Borel $\sigma$-algebra of $[0,T]\times R_k(T)$, so $h^{A,j}_k$ is measurable. Hence $h^A$ is an observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels.

Claim 2. Let $(\sigma^i,\Upsilon^\upsilon,\alpha)$ be a solution with regular event $\Omega_0$, and let $\omega\in\Omega_0$ and $t\in[0,T]$. The control identity, condition 5 of the definition of a solution, states that

where $K_t$ is the number of observation events up to time $t$ and the $\tau_j,\upsilon_j$ are the corresponding jump times and channels, the right-hand side being $h^A_0(t)$ when $K_t=0$. By the definition of $h^A$, the right-hand side equals $A_t$ for every value of $K_t$ and every choice of the remaining arguments. Hence $\alpha_t(\omega)=A_t$. $\blacksquare$