Proof of 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

Throughout, $\mathcal{B}_{[0,a]}$ denotes the trace Borel $\sigma$-algebra on $[0,a]$ for $a>0$, and we use that the family of subsets of a set whose preimage under a given map lies in a given $\sigma$-algebra is a $\sigma$-algebra, so that a map into a measurable space whose $\sigma$-algebra is generated by a family $\mathcal{C}$ of sets is measurable as soon as the preimages of the members of $\mathcal{C}$ are measurable, and that compositions of measurable maps are measurable. The empty record is $r_\emptyset=(0,(),())$. We also use the following fact about a counting path $c$ and its jump times $\tau_j(c)$: for every $t\ge0$ and every natural number $j$, $\tau_j(c)\le t$ if and only if $c(t)\ge j$. Indeed, if $c(t)\ge j$ then $t$ belongs to the set whose greatest lower bound is $\tau_j(c)$; conversely, if $\tau_j(c)\le t$, then the set $\{r'\ge0:c(r')\ge j\}$ is nonempty and for every $u>\tau_j(c)$ contains some $r'<u$, so $c(u)\ge c(r')\ge j$ for all $u>\tau_j(c)$ by monotonicity, so $c(\tau_j(c))\ge j$ by right-continuity (condition 3 of Counting Path and Its Jump Times), and $c(t)\ge c(\tau_j(c))\ge j$. Moreover $c(\tau_j(c))=j$ whenever $\tau_j(c)<\infty$, each such $\tau_j(c)$ is a jump time, $\tau_1(c)<\tau_2(c)<\dots$ as long as they are finite, and every jump time $u$ of $c$ equals $\tau_{c(u)}(c)$; hence for every $t\ge0$ the jump times of $c$ in $(0,t]$ are exactly $\tau_1(c)<\dots<\tau_{c(t)}(c)$. Indeed, $c(r')<j$ for $r'<\tau_j(c)$ by the fact just proved, so $c(\tau_j(c)-)\le j-1$ by integrality, and unit jumps (condition 4 of Counting Path and Its Jump Times) give $c(\tau_j(c))\le j$, while $c(\tau_j(c))\ge j$ by the fact; so $c(\tau_j(c))=j>c(\tau_j(c)-)$ (and $\tau_j(c)>0$, since $c(0)=0<j$), and $\tau_{j+1}(c)>\tau_j(c)$ because $c(\tau_j(c))=j<j+1$ and $c(r')<j+1$ for $r'<\tau_j(c)$. Conversely, if $c(u)>c(u-)$ with $u>0$, put $j=c(u)$: then $\tau_j(c)\le u$ by the fact, and $c(r')\le c(u-)=j-1<j$ for $r'<u$ by unit jumps and integrality, so $\tau_j(c)\ge u$. Finally, a jump time $u\le t$ has $c(u)\le c(t)$, and $\tau_j(c)\le t$ for $j\le c(t)$ by the fact. (A jump time $u$ of $c$ is a property of the values of $c$ on $[0,u]$ alone.)

Claim 1. Fix $k\ge1$, a mark vector $v$ and a component index $j$. The map $(t,\tau)\mapsto h^j_k(t,\tau,v)$ on $[0,T]\times R_k(T)$ is measurable with respect to the $\sigma$-algebra $\mathcal{S}_T$ generated by the sets $U\cap([0,T]\times R_k(T))$ with $U$ open in $\mathbb{R}^{1+k}$. The inclusion map $\mathfrak{j}:[0,s]\times R_k(s)\to[0,T]\times R_k(T)$ is measurable with respect to the $\sigma$-algebra $\mathcal{S}_s$ generated by the sets $U\cap([0,s]\times R_k(s))$ and $\mathcal{S}_T$: the preimage of a generator $U\cap([0,T]\times R_k(T))$ of $\mathcal{S}_T$ is $U\cap([0,s]\times R_k(s))$, a generator of $\mathcal{S}_s$, because $[0,s]\times R_k(s)\subseteq[0,T]\times R_k(T)$. Hence $(t,\tau)\mapsto h^{(s),j}_k(t,\tau,v)=h^j_k(\mathfrak{j}(t,\tau),v)$ is $\mathcal{S}_s$-measurable. The same argument with the inclusion $[0,s]\to[0,T]$ handles $h^{(s)}_0$. Thus $h^{(s)}$ satisfies the measurability requirement of Observation-Driven Control Policy with horizon $s$; it is $\mathcal{A}$-valued because its values are values of $h$, which is $\mathcal{A}$-valued.

Claim 2. We write $\mathcal{T}^{(s)}$, $N^{(s)}$, and so on, for the derived quantities of the restricted families, formed as in Solution of the Controlled N-Agent Dynamics with horizon $s$ and regular event $\Omega_0$, and verify the conditions of that definition. Throughout, $\omega\in\Omega_0$ is fixed for the pathwise conditions 1, 3, 4, 5, 6.

Condition 1. $\sigma^i_0=\varsigma^i_0$ is inherited. The path $t\mapsto\sigma^i_t$ is constant on $[0,t^{(i)}_1)$, on each $[t^{(i)}_j,t^{(i)}_{j+1})$ and on $[t^{(i)}_{K^{(i)}},T]$ for times $0<t^{(i)}_1<\dots<t^{(i)}_{K^{(i)}}\le T$. Let $k'$ be the number of indices $j$ with $t^{(i)}_j\le s$. The restriction to $[0,s]$ is constant on $[0,t^{(i)}_1)$ if $k'\ge1$, on $[t^{(i)}_j,t^{(i)}_{j+1})$ for $j<k'$, and on $[t^{(i)}_{k'},s]$, the latter being contained in $[t^{(i)}_{k'},t^{(i)}_{k'+1})$ if $k'<K^{(i)}$ and in $[t^{(i)}_{K^{(i)}},T]$ if $k'=K^{(i)}$; if $k'=0$ it is constant on $[0,s]\subseteq[0,t^{(i)}_1)$ (or on $[0,s]\subseteq[0,T]$ if $K^{(i)}=0$). So condition 1 holds on $[0,s]$ with the times $t^{(i)}_1,\dots,t^{(i)}_{k'}$.

Condition 2. The occupation indicators and the empirical state measure of the restricted families are, by the derived notation, the restrictions to $t\in[0,s]$ of $\eta^{i,\gamma}_t$ and $\Sigma_t$; in particular $\Sigma^{(s)}_t=\Sigma_t$ for $t\le s$. The two maps required to be measurable on $[0,s]\times\Omega$ are the restrictions to $[0,s]\times\Omega$ of the corresponding maps on $[0,T]\times\Omega$, which are $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable by condition 2 for the given solution. The inclusion $[0,s]\times\Omega\to[0,T]\times\Omega$ is measurable with respect to the product $\sigma$-algebras $\mathcal{B}_{[0,s]}\otimes\mathcal{F}$ and $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$; by that definition the latter is generated by the measurable rectangles, so by the preamble it suffices that the preimage of a measurable rectangle $A\times C$ with $A\in\mathcal{B}_{[0,T]}$ and $C\in\mathcal{F}$ is $(A\cap[0,s])\times C$, and $A\cap[0,s]\in\mathcal{B}_{[0,s]}$ because $A=S\cap[0,T]$ for a Borel set $S$ and $S\cap[0,T]\cap[0,s]=S\cap[0,s]$. Hence the restricted maps are $\mathcal{B}_{[0,s]}\otimes\mathcal{F}$-measurable. For $t\in[0,s]$ the consumed clock times of the restricted families are the Lebesgue integrals over the compact interval $[0,t]$ of the same integrands as those of the given solution (the integral over $[0,t]$ depends only on the integrand on $[0,t]$ and on $\mathcal{B}_{[0,t]}$, whichever ambient interval is used), so $\mathcal{T}^{(s),i,\sigma\gamma}_t=\mathcal{T}^{i,\sigma\gamma}_t$ and $\tilde{\mathcal{T}}^{(s),i,\upsilon}_t=\tilde{\mathcal{T}}^{i,\upsilon}_t$ for all $t\in[0,s]$ and every $\omega\in\Omega$.

Condition 3. Consequently the counters of the restricted families are $N^{(s),i,\sigma\gamma}_t=Y^{i,\sigma\gamma}_{\mathcal{T}^{i,\sigma\gamma}_t}=N^{i,\sigma\gamma}_t$ and $\tilde{N}^{(s),i,\upsilon}_t=\tilde{N}^{i,\upsilon}_t$ for $t\in[0,s]$, and likewise the observation total and the grand total are the restrictions of $\tilde{c}_t$ and of the grand total of the given solution. Each of these coincides on $[0,T]$, hence on $[0,s]$, with the restriction of a counting path, which is condition 3 on $[0,s]$.

Condition 4 holds for $t\in[0,s]$ because it holds for $t\in[0,T]$ and $\Upsilon^{\upsilon}_t$, $\tilde{N}^{i,\upsilon}_t$ are unchanged.

Condition 5. The observation-event count of the restricted families is $K^{(s)}_t=\tilde{c}_t=K_t$ for $t\in[0,s]$. Let $c$ be a counting path whose restriction to $[0,T]$ is $t\mapsto\tilde{c}_t$, so that, by the preamble, the jump times of $c$ in $(0,T]$ are $\tau_1(c)<\dots<\tau_{c(T)}(c)$ with $c(T)=K_T$; since $\tau_1<\dots<\tau_{K_T}$ is the increasing list of these same jump times, $\tau_j=\tau_j(c)$ for $j\le K_T$. The jump times of the observation total of the restricted families in $[0,s]$ are the jump times $u\in(0,s]$ of $c$ (a jump time being a property of the values of $c$ on $[0,u]$ alone), which by the preamble are $\tau_1(c)<\dots<\tau_{c(s)}(c)$ with $c(s)=K_s$. So the observation event times of the restricted families are $\tau_1<\dots<\tau_{K_s}$, and their channels are $\upsilon_1,\dots,\upsilon_{K_s}$, each $\upsilon_j$ being the unique channel such that some observation counter with that channel jumps at $\tau_j$, a property of the counters on $[0,\tau_j]\subseteq[0,s]$. For $t\in[0,s]$, condition 5 for the given solution reads $\alpha_t=h_{K_t}(t,\tau_1,\dots,\tau_{K_t},\upsilon_1,\dots,\upsilon_{K_t})$; here $K_t\le K_s$, the tuple $(\tau_1,\dots,\tau_{K_t})$ lies in $R_{K_t}(s)$ since $\tau_j\le t\le s$ for $j\le K_t$ (again by the preamble fact), and $t\in[0,s]$; so the right side equals $h^{(s)}_{K_t}(t,\tau_1,\dots,\tau_{K_t},\upsilon_1,\dots,\upsilon_{K_t})$ by the definition of $h^{(s)}$ (read as $h^{(s)}_0(t)=h_0(t)$ when $K_t=0$). This is condition 5 on $[0,s]$ for the policy $h^{(s)}$.

Condition 6 holds for $t\in[0,s]$ because it holds on $[0,T]$ with the same counters and indicators.

Thus the restricted families with the regular event $\Omega_0$ form a solution on $[0,s]$ for $h^{(s)}$, and the identities for the derived quantities have been established along the way; the empirical state measure agrees for every $t\in[0,s]$ and every $\omega$ by the derived notation. The observation filtration of the restricted solution at $t\in[0,s]$ is generated by the random variables $\Upsilon^\upsilon_u$ with $u\le t$ together with every event of $\mathcal{F}$ of probability zero (the same $\mathcal{F}$ and $P$, hence the same null events), which is exactly $\mathcal{G}_t$; likewise the system filtration at $t\in[0,s]$ is generated by the initial states, the counters $N^{i,\sigma\gamma}_u$, $\tilde{N}^{i,\upsilon}_u$ with $u\le t$ (equal to those of the given solution) and the null events, which is $\mathcal{F}^{\mathrm{sys}}_t$.

The record. By The Observation Record of a Solution of the Controlled N-Agent Dynamics applied to the restricted solution, $W^{(s)}(\omega)=(K_s,(\tau_1,\dots,\tau_{K_s}),(\upsilon_1,\dots,\upsilon_{K_s}))$ for $\omega\in\Omega_0$ with $K_s(\omega)\ge1$, and $W^{(s)}(\omega)=r_\emptyset$ otherwise. On the other hand, for $\omega\in\Omega_0$ with $K_T(\omega)\ge1$, $W(\omega)=(K_T,(\tau_1,\dots,\tau_{K_T}),(\upsilon_1,\dots,\upsilon_{K_T}))$ and, by the definition of the prefix map, $\pi_s(W(\omega))=(\kappa,(\tau_1,\dots,\tau_\kappa),(\upsilon_1,\dots,\upsilon_\kappa))$ with $\kappa$ the number of indices $j\le K_T$ with $\tau_j\le s$, which is $K_s$ by the preamble fact ($\tau_j\le s$ if and only if $c(s)\ge j$); this is $W^{(s)}(\omega)$ when $K_s\ge1$, and, the blocks being empty, is $(0,(),())=r_\emptyset=W^{(s)}(\omega)$ when $K_s=0$. If $\omega\in\Omega_0$ with $K_T(\omega)=0$, or $\omega\notin\Omega_0$, then $W(\omega)=r_\emptyset$, $\pi_s(r_\emptyset)=r_\emptyset$ (the empty record has no event times, so $\kappa=0$ and the blocks are empty), and $W^{(s)}(\omega)=r_\emptyset$ (in the first case $K_s\le K_T=0$). Hence $W^{(s)}=\pi_s\circ W$ on all of $\Omega$.