The Open-Loop Policy Determined by a Measurable Control

Let $T>0$ be a real number, let $m$ and $\tilde{l}$ be natural numbers, and let $A:[0,T]\to\mathbb{R}^m$ be a map whose components are measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line. Define a family $h^A=(h^A_k)_{k\ge0}$ by

for $t\in[0,T]$, ordered times $0\le\tau_1\le\dots\le\tau_k\le T$ and channels $\upsilon_1,\dots,\upsilon_k\in\{1,\dots,\tilde{l}\}$.

1. $h^A$ is an observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels; it is called the open-loop policy determined by $A$.

2. In the setting of the controlled $N$-agent dynamics with policy $h^A$, every solution on $[0,T]$ satisfies $\alpha_t(\omega)=A_t$ for every $t\in[0,T]$ and every $\omega$ in the regular event.