Weighted Compensated Sums over the Observation Events of the Controlled N-Agent Dynamics

Adopt the setting of the \reftext{lem:n-agent-compensated-martingales-2026a}{compensated counters of the controlled $N$-agent dynamics}: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with rate bound $\tilde{B}$ and $\tilde{l}$ channels, a horizon $T>0$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} $(\Omega,\mathcal{F},P)$, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h$, and a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} on $[0,T]$ with regular event $\Omega_0$, state processes $\sigma^i$, empirical state measure $\Sigma_t$, observation counters $\tilde{N}^{i,\upsilon}_t$, consumed clock times $\tilde{A}^{i,\upsilon}_t$, observation total $\tilde{c}_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$. Write $M^{(i,\upsilon)}_t=\tilde{N}^{i,\upsilon}_t-\tilde{A}^{i,\upsilon}_t$ for the compensated counters of the \reftext{lem:n-agent-compensated-martingales-2026a}{compensated-counters lemma} attached to the \textbf{observation clock labels} $a=(i,\upsilon)$, $i\in\{1,\dots,N\}$, $\upsilon\in\{1,\dots,\tilde{l}\}$; at every $\omega\in\Omega_0$ let $\tau_1<\dots<\tau_{\tilde{c}_T}$ be the jump times of the observation total in $[0,T]$ and $\upsilon_1,\dots,\upsilon_{\tilde{c}_T}$ their channels, as in condition 5 of the \reftext{def:n-agent-controlled-dynamics-2026a}{solution definition} (where $\tilde{c}_t$ is written $K_t$). Let $\tilde{b}$ be the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$, with $\tilde{b}(\Sigma)$ the vector with components $\tilde{b}^1(\Sigma),\dots,\tilde{b}^{\tilde{l}}(\Sigma)$, and let $M^\gamma_t$ ($\gamma\in\{1,\dots,l\}$) be the state martingales of the \reftext{thm:n-agent-martingale-decomposition-2026a}{martingale decomposition}. Write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation}, \reftext{def:square-integrable-mean-square-2026a}{square-integrable} for membership in the mean-square space, \reftext{def:lebesgue-integral-integrable-2026a}{integrable} for finiteness of the expectation of the absolute value, $\mathbf{1}_D$ for the function equal to $1$ on a set $D$ and $0$ off $D$, $e_\upsilon$ for the $\upsilon$-th standard basis vector of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^{\tilde{l}}$, and $\sqrt{N}=N^{1/2}$ for the \reftext{thm:nonnegative-real-has-unique-square-root-2026a}{nonnegative square root}; vectors are identified with one-column matrices and products of matrices are \reftext{def:product-real-matrices-2026a}{matrix products}, with $(\cdot)^{\top}$ the \reftext{def:transpose-real-matrix-2026a}{transpose}.

Fix a natural number $d\ge1$ and an assignment $F$ of a real matrix $F_t$ with $d$ rows and $\tilde{l}$ columns to each $t\in[0,T]$ such that every entry map $t\mapsto F^{\gamma\upsilon}_t$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$, and fix a real $\bar{F}\ge0$ with $|F^{\gamma\upsilon}_t|\le\bar{F}$ for all $\gamma$, $\upsilon$, $t$ (such a bound exists because continuous functions on a compact interval are \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}). Let likewise $G$ be an assignment of real matrices with $d'$ rows and $\tilde{l}$ columns with continuous entries and entry bound $\bar{G}$. Define, for $t\in[0,T]$ and at every $\omega\in\Omega$, the \textbf{weighted observation sum} and the \textbf{weighted compensated observation sum}

where the sum is $0$ when $\tilde{c}_t=0$ (in particular off $\Omega_0$, where all counters vanish), and the integral is the componentwise \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,t]$ of an integrand with values bounded by $\bar{F}\,\tilde{l}\,\tilde{B}$ in absolute value whose sections in $r$ are measurable by conclusion (a) (in particular the integrand, and hence the integral, is well defined and equal to $0$ at every $\omega\notin\Omega_0$, where the factor $\mathbf{1}_{\Omega_0}$ vanishes identically), the integral being $0$ for $t=0$.

\textbf{(a) (Representation and regularity.)} At every $\omega\in\Omega_0$: for each $\upsilon$, the jump times in $[0,T]$ of the path $t\mapsto\sum_{i=1}^{N}\tilde{N}^{i,\upsilon}_t$ are exactly the $\tau_j$ with $\upsilon_j=\upsilon$, each jump having size $1$; the identity

holds for every $t\in[0,T]$ and every $\upsilon$; and for all $0\le r\le t\le T$ and every $\gamma\in\{1,\dots,d\}$,

and every path $t\mapsto\tilde{J}^{F,\gamma}_t$ is right-continuous on $[0,T]$ with $\tilde{J}^{F}_0=0$. Moreover, for every $\upsilon\in\{1,\dots,\tilde{l}\}$, the map $s\mapsto\mathbf{1}_{\Omega_0}\tilde{b}^\upsilon(\Sigma_s)$ is a section of a map that is \reftext{def:measurable-function-2026a}{measurable} for the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} of the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and $\mathcal{F}$; for every $t\in[0,T]$ and $\gamma\in\{1,\dots,d\}$, $\tilde{J}^{F,\gamma}_t$ is a \reftext{def:probability-space-random-variable-2026a}{random variable} that is measurable with respect to $\mathcal{F}^{\mathrm{sys}}_t$, with $\mathbb{E}\big[|\tilde{J}^{F,\gamma}_t|^{p}\big]<\infty$ for every natural number $p\ge1$; and for every $\gamma\in\{1,\dots,d\}$, the map $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\tilde{J}^{F,\gamma}_t(\omega)$ is product-measurable.

\textbf{(b) (First-order identity.)} Let $r\in[0,T]$ and let $Z$ be an $\mathcal{F}^{\mathrm{sys}}_r$-measurable square-integrable random variable. Then for every $t\in[r,T]$ and every $\gamma\in\{1,\dots,d\}$ the product $Z\,(\tilde{J}^{F,\gamma}_t-\tilde{J}^{F,\gamma}_r)$ is integrable and

In particular (taking $Z=\mathbf{1}_D$, $D\in\mathcal{F}^{\mathrm{sys}}_r$) each component $(\tilde{J}^{F,\gamma}_t)_{t\in[0,T]}$ is a \reftext{def:square-integrable-martingale-2026a}{square-integrable martingale} with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ (time index restricted to $[0,T]$), vanishing at $0$ almost surely.

\textbf{(c) (Second-order identity.)} Let $r\in[0,T]$ and let $Z$ be an $\mathcal{F}^{\mathrm{sys}}_r$-measurable random variable whose square $Z^2$ is square-integrable. Then for every $t\in[r,T]$, every $\gamma\in\{1,\dots,d\}$, and every $\delta\in\{1,\dots,d'\}$, the products below are integrable and

the inner integral existing at every $\omega$ as the Lebesgue integral over $[r,t]$ of a measurable function bounded in absolute value by $\bar{F}\bar{G}\tilde{l}\tilde{B}$. In particular, with $Z=1$ and $r=0$: writing $D(\Sigma)$ for the diagonal matrix with $\tilde{l}$ rows and columns and diagonal entries $\tilde{b}^1(\Sigma),\dots,\tilde{b}^{\tilde{l}}(\Sigma)$,

and, specializing $G=F$, $d'=d$, and $\delta=\gamma$,

\textbf{(d) (Cross identity with the state martingales.)} Let $r$ and $Z$ be as in (c). Then for every $t\in[r,T]$, every $\gamma\in\{1,\dots,l\}$, and every $\delta\in\{1,\dots,d\}$, the product below is integrable and

\textbf{(e) (Fourth-moment bound.)} For every $t\in[0,T]$ and every $\gamma\in\{1,\dots,d\}$,

in particular the bound does not depend on $N$, on the driving system, or on the solution.