Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record

Adopt the setting of the controlled $N$-agent dynamics: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, a nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$ in Euclidean space, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}$, a real number $T>0$ (the horizon), an $N$-agent driving system $(\Omega,\mathcal{F},P)$ with initial states $\varsigma^i_0$, transition clocks $Y^{i,\sigma\gamma}$ and observation clocks $\tilde{Y}^{i,\upsilon}$, and an $\mathcal{A}$-valued observation-driven control policy $h=(h_k)_{k\ge0}$ with horizon $T$, control dimension $m$ and $\tilde{l}$ channels, with the record spaces $R_k(T)$ of that definition. Let $s$ be a real number with $0<s<T$ (the letter $s$ is the fixed restriction time throughout; the integration variable written $s$ in condition 2 of Solution of the Controlled N-Agent Dynamics is written $u$ here), and let $\pi_s:\mathbf{R}(T,\tilde{l})\to\mathbf{R}(s,\tilde{l})$ be the prefix map between the observation record spaces with horizons $T$ and $s$.

The truncated policy $h^{(s)}=(h^{(s)}_k)_{k\ge0}$ is defined by $h^{(s)}_0(t)=h_0(t)$ for $t\in[0,s]$ and $h^{(s)}_k(t,\tau,v)=h_k(t,\tau,v)$ for $k\ge1$, $t\in[0,s]$, $\tau\in R_k(s)$ and $v\in\{1,\dots,\tilde{l}\}^k$ (note $R_k(s)\subseteq R_k(T)$).

1. (The truncated policy.) $h^{(s)}$ is an observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels, and it is $\mathcal{A}$-valued.

2. (The restricted solution.) Let state processes $\sigma^i$, observation processes $\Upsilon^\upsilon$, a control process $\alpha$ and a regular event $\Omega_0$ form a solution of the controlled $N$-agent dynamics on $[0,T]$ for the policy $h$, with consumed clock times $\mathcal{T}^{i,\sigma\gamma}_t$, $\tilde{\mathcal{T}}^{i,\upsilon}_t$, counters $N^{i,\sigma\gamma}_t$, $\tilde{N}^{i,\upsilon}_t$, observation total $\tilde{c}_t$, observation-event count $K_t$, observation event times $\tau_1<\dots<\tau_{K_T}$ and channels $\upsilon_1,\dots,\upsilon_{K_T}$, empirical state measure $\Sigma_t$, observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$, system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and observation record $W$. Then the families $(\sigma^i_t)_{t\in[0,s]}$, $(\Upsilon^\upsilon_t)_{t\in[0,s]}$ and $(\alpha_t)_{t\in[0,s]}$, together with the same regular event $\Omega_0$, form a solution of the controlled $N$-agent dynamics on $[0,s]$ for the policy $h^{(s)}$ (the same driving system, rate families and control set, with the horizon $s$ in place of $T$). Its consumed clock times, counters, observation total, observation-event count, empirical state measure, observation filtration and system filtration are the restrictions to $t\in[0,s]$ of those of the given solution: for every $t\in[0,s]$, every $\omega\in\Omega$ and all indices they equal $\mathcal{T}^{i,\sigma\gamma}_t$, $\tilde{\mathcal{T}}^{i,\upsilon}_t$, $N^{i,\sigma\gamma}_t$, $\tilde{N}^{i,\upsilon}_t$, $\tilde{c}_t$, $K_t$ and $\Sigma_t$, and its observation and system filtrations satisfy $\mathcal{G}^{(s)}_t=\mathcal{G}_t$ and $\mathcal{F}^{\mathrm{sys},(s)}_t=\mathcal{F}^{\mathrm{sys}}_t$ for every $t\in[0,s]$; at every $\omega\in\Omega_0$ its observation event times and channels are $\tau_1<\dots<\tau_{K_s}$ and $\upsilon_1,\dots,\upsilon_{K_s}$ (the empty lists when $K_s(\omega)=0$); and its observation record $W^{(s)}:\Omega\to\mathbf{R}(s,\tilde{l})$ satisfies

$$W^{(s)}(\omega)=\pi_s\bigl(W(\omega)\bigr)\qquad\text{for every }\omega\in\Omega .$$