Multiplier Identities and Interval Estimates for the Compensated Counters 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 clock labels $a$, counters $N^a_t$, consumed clock times $A^a_t$, compensated counters $M^a_t=N^a_t-A^a_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, all as in that lemma. Let $n_c=Nl(l-1)+N\tilde{l}$ be the number of clock labels, let $\bar{\beta}=\max(B,\tilde{B})$, 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, and \reftext{def:lebesgue-integral-integrable-2026a}{integrable} for finiteness of the expectation of the absolute value, and set

Fix $r\in[0,T]$ and a \reftext{def:probability-space-random-variable-2026a}{random variable} $Z$ on $(\Omega,\mathcal{F},P)$ that is \reftext{def:measurable-function-2026a}{measurable} with respect to $\mathcal{F}^{\mathrm{sys}}_r$.

\textbf{(a) (First-order identity.)} If $Z$ is square-integrable, then for every clock label $a$ and every $t\in[r,T]$ the product $Z\,(M^a_t-M^a_r)$ is integrable and

\textbf{(b) (Second-order identity.)} If $Z$, $Z\,M^a_r$, and $Z\,M^b_r$ are square-integrable, where $a$ and $b$ are clock labels, then for every $t\in[r,T]$ the products $Z\,(M^a_t-M^a_r)(M^b_t-M^b_r)$ and $Z\,(A^a_t-A^a_r)$ are integrable and

\textbf{(c) (Moments of every order.)} Let $\Xi_t=\sum_a N^a_t$, the sum over all clock labels $a$ (there are $n_c$ of them). For every \reftext{def:natural-numbers-2026a}{natural number} $p\ge1$ and every $t\in[0,T]$,

and consequently every finite product $M^{a_1}_{t_1}\cdots M^{a_k}_{t_k}$ (clock labels $a_1,\dots,a_k$ and times $t_1,\dots,t_k\in[0,T]$ arbitrary, $k\ge1$) is integrable.

In parts (d), (e), (f), fix $\delta$ with $0<\delta\le T-r$, write $\Delta M^a=M^a_{r+\delta}-M^a_r$ and $\Delta A^a=A^a_{r+\delta}-A^a_r$ for clock labels $a$, and let $k\in\{3,4\}$.

\textbf{(d) (Pure powers, bounded multiplier.)} If $|Z|\le\zeta$ everywhere for a real number $\zeta\ge0$, then for every clock label $a$ the product $Z\,((\Delta M^a)^k-\Delta A^a)$ is integrable and

\textbf{(e) (Pure powers, square-integrable multiplier.)} If $Z$ is square-integrable, then for every clock label $a$ the product $Z\,((\Delta M^a)^k-\Delta A^a)$ is integrable and, with $\sqrt{\delta}$ the \reftext{thm:nonnegative-real-has-unique-square-root-2026a}{nonnegative square root},

\textbf{(f) (Mixed products.)} If $Z$ is integrable and $a_1,\dots,a_j$ are clock labels with $2\le j\le4$, \textbf{not all equal}, then the product $|Z|\,\prod_{q=1}^{j}|\Delta M^{a_q}|$ (written with the \reftext{def:finite-product-notation-2026a}{finite product notation}) is integrable and