The Record-Frozen Control Path and Record-Frozen Policy

Let $m\ge1$ and $\tilde{l}\ge1$ be natural numbers, let $T>0$ be a real number, let $h=(h_k)_{k\ge0}$ be an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, and let $r$ be a member of the observation record space $\mathbf{R}(T,\tilde{l})$, written $r=(k,t,v)$ with $t=(t_1,\dots,t_k)$ and $v=(v_1,\dots,v_k)$, where $k=0$ for the empty record.

For $s\in[0,T]$ let $k_r(s)$ denote the number of indices $j\in\{1,\dots,k\}$ with $t_j\le s$ (so an event at time $s$ is counted, matching $K_s$ in condition 5 of Solution of the Controlled N-Agent Dynamics, the observation total there being right-continuous). The record-frozen control path of $h$ at $r$ is the map $a^r:[0,T]\to\mathbb{R}^m$ given by $a^r(s)=h_{k_r(s)}\bigl(s,\ (t_1,\dots,t_{k_r(s)}),\ (v_1,\dots,v_{k_r(s)})\bigr),$ read as $h_0(s)$ when $k_r(s)=0$; this is well defined since $(t_1,\dots,t_j)$ lies in the set $R_j(T)$ of Observation-Driven Control Policy for every $j\in\{1,\dots,k\}$.

Each component of $a^r$ is measurable on $[0,T]$ with the $\sigma$-algebra required by Observation-Driven Control Policy, which coincides with the trace Borel $\sigma$-algebra: the traces of Borel sets form a $\sigma$-algebra containing the relatively open subsets of $[0,T]$, while the Borel sets whose traces lie in the $\sigma$-algebra generated by the relatively open sets form a $\sigma$-algebra containing the open sets, so the two $\sigma$-algebras agree. Measurability holds because, for fixed $j$ and fixed record arguments $(\tau,w)$, the map $s\mapsto(s,\tau)$ from $[0,T]$ to $[0,T]\times R_j(T)$ is continuous, so preimages of relatively open sets are relatively open and the composition $s\mapsto h_j(s,\tau,w)$ is measurable; and $a^r$ agrees with finitely many such compositions on the members of the measurable partition of $[0,T]$ into the intervals on which $k_r$ is constant.

The record-frozen policy at $r$ is the observation-driven control policy $\hat{h}^r=((\hat{h}^r)_j)_{j\ge0}$ with horizon $T$, control dimension $m$, and $\tilde{l}$ channels whose members are constant in the record arguments, with values given by the record-frozen control path: $(\hat{h}^r)_0(s)=a^r(s),\qquad (\hat{h}^r)_j(s,\tau,w)=a^r(s)$ for every $j\ge1$, $s\in[0,T]$, $\tau\in R_j(T)$, and $w\in\{1,\dots,\tilde{l}\}^j$. Each $(\hat{h}^r)_j$ satisfies the measurability required in Observation-Driven Control Policy: its preimages are the sets $E\times R_j(T)$ with $E$ a trace Borel subset of $[0,T]$, and writing $E=S\cap[0,T]$ with $S$ Borel in $\mathbb{R}$ gives $E\times R_j(T)=(S\times\mathbb{R}^{j})\cap([0,T]\times R_j(T))$, so these sets lie in the $\sigma$-algebra generated by the relatively open subsets of $[0,T]\times R_j(T)$, by the same two-inclusion argument as above applied to $[0,T]\times R_j(T)$.