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

Adopt the setting of the compensated counters of the controlled $N$-agent dynamics: a transition-rate family $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$ and $\tilde{l}$ channels, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$, and a 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 $\tilde{\mathcal{T}}^{i,\upsilon}_t$, observation total $\tilde{c}_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$. Write $M^{(i,\upsilon)}_t=\tilde{N}^{i,\upsilon}_t-\tilde{\mathcal{T}}^{i,\upsilon}_t$ for the compensated counters of the compensated-counters lemma attached to the 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 solution definition (where $\tilde{c}_t$ is written $K_t$). Let $\tilde{b}$ be the 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 martingale decomposition. Write $\mathbb{E}$ for the expectation, square-integrable for membership in the mean-square space, 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 Euclidean space $\mathbb{R}^{\tilde{l}}$, and $\sqrt{N}=N^{1/2}$ for the nonnegative square root; vectors are identified with one-column matrices and products of matrices are matrix products, with $(\cdot)^{\top}$ the transpose.

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. 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 $t\mapsto F^{p\upsilon}_t$ is continuous on $[0,T]$, and fix a real $\bar{F}\ge0$ with $|F^{p\upsilon}_t|\le\bar{F}$ for all $p$, $\upsilon$, $t$. Fix likewise a natural number $d'\ge1$ and an assignment $G$ of a real matrix $G_t$ with $d'$ rows and $\tilde{l}$ columns to each $t\in[0,T]$, every entry map $t\mapsto G^{q\upsilon}_t$ continuous on $[0,T]$, with a real $\bar{G}\ge0$ satisfying $|G^{q\upsilon}_t|\le\bar{G}$ for all $q$, $\upsilon$, $t$. Define, for $t\in[0,T]$ and at every $\omega\in\Omega$, the weighted observation sum and the weighted compensated observation sum

where the sum is $0$ when $\tilde{c}_t=0$ (in particular off $\Omega_0$, where all counters vanish by part (vii)(c) of the existence theorem), and the integral is the componentwise Lebesgue integral over the compact interval $[0,t]$ of an integrand with values bounded by $\bar{F}\,\tilde{l}\,\tilde{B}$ in absolute value, componentwise progressively measurable by conclusion (a) together with claim 3 of the progressive-measurability toolkit (the integrand, and hence the integral, vanishes at every $\omega\notin\Omega_0$, where the factor $\mathbf{1}_{\Omega_0}$ vanishes identically), the integral being $0$ for $t=0$.

(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 $p\in\{1,\dots,d\}$,

and, at every $\omega\in\Omega$, the path $t\mapsto\tilde{J}^{F,p}_t$ is right-continuous on $[0,T]$, with $\tilde{J}^{F}_0=0$ everywhere and $\tilde{J}^{F}\equiv0$ off $\Omega_0$. Moreover: for every $\upsilon\in\{1,\dots,\tilde{l}\}$ the family $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}\tilde{b}^\upsilon(\Sigma_s)$, $s\in[0,T]$, is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$; and for every $p\in\{1,\dots,d\}$ the component family $(\tilde{J}^{F,p}_t)_{t\in[0,T]}$ is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ — in particular each $\tilde{J}^{F,p}_t$ is an $\mathcal{F}^{\mathrm{sys}}_t$-measurable random variable and $(t,\omega)\mapsto\tilde{J}^{F,p}_t(\omega)$ is measurable for the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$ — with $\mathbb{E}\big[|\tilde{J}^{F,p}_t|^{k}\big]<\infty$ for every natural number $k\ge1$.

(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 $p\in\{1,\dots,d\}$ the product $Z\,(\tilde{J}^{F,p}_t-\tilde{J}^{F,p}_r)$ is integrable and

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

(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 $p\in\{1,\dots,d\}$, and every $q\in\{1,\dots,d'\}$, 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 $\bar{F}\bar{G}\tilde{l}\tilde{B}$, and read as $0$ when $t=r$. 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 $q=p$,

(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 $p\in\{1,\dots,d\}$, the product below is integrable and

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

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