Multiplier Identities and Interval Estimates for the Compensated Counters of the Controlled N-Agent Dynamics

Adopt the setting of the compensated counters of the controlled $N$-agent dynamics: a nonempty subset $\mathcal{A}$ of Euclidean space $\mathbb{R}^m$, a transition-rate family $\beta$ with rate bound $B$ on $l$ states with control set $\mathcal{A}$, 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 $h$ which is $\mathcal{A}$-valued, and a solution on $[0,T]$, with clock labels $a$, counters $N^a_t$, consumed clock times $\mathcal{T}^a_t$, compensated counters $M^a_t=N^a_t-\mathcal{T}^a_t$, 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 expectation, square-integrable for membership in the mean-square space, and integrable for finiteness of the expectation of the absolute value, and set

$$C_\star=2^{20}\,(n_c+1)\,(1+\bar{\beta})^2\,(1+\bar{\beta}T)^4 .$$

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

(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

$$\mathbb{E}\big[Z\,(M^a_t-M^a_r)\big]=0 .$$

(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 $Z\,(\mathcal{T}^a_t-\mathcal{T}^a_r)$ are integrable and

$$\mathbb{E}\big[Z\,(M^a_t-M^a_r)(M^b_t-M^b_r)\big]=\begin{cases}\mathbb{E}\big[Z\,(\mathcal{T}^a_t-\mathcal{T}^a_r)\big]&\text{if }a=b,\\ 0&\text{if }a\neq b.\end{cases}$$

(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 natural number $p\ge1$ and every $t\in[0,T]$,

$$\mathbb{E}\big[\Xi_t^{\,p}\big]\ \le\ n_c^{\,p+1}\,\big((2p)^p+2^p\,(\bar{\beta}t)^p\big)\ <\ \infty ,$$

and consequently every finite product $M^{a_1}_{t_1}\cdots M^{a_\kappa}_{t_\kappa}$ (clock labels $a_1,\dots,a_\kappa$ and times $t_1,\dots,t_\kappa\in[0,T]$ arbitrary, $\kappa\ge1$) 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 $\Delta\mathcal{T}^a=\mathcal{T}^a_{r+\delta}-\mathcal{T}^a_r$ for clock labels $a$; in parts (d) and (e) let moreover $k\in\{3,4\}$.

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

$$\Big|\mathbb{E}\big[Z\,\big((\Delta M^a)^k-\Delta\mathcal{T}^a\big)\big]\Big|\ \le\ C_\star\,\zeta\,\delta^2 .$$

(e) (Pure powers, square-integrable multiplier.) If $Z$ is square-integrable, then for every clock label $a$ the product $Z\,((\Delta M^a)^k-\Delta\mathcal{T}^a)$ is integrable and, with $\sqrt{\delta}$ the nonnegative square root,

$$\Big|\mathbb{E}\big[Z\,\big((\Delta M^a)^k-\Delta\mathcal{T}^a\big)\big]\Big|\ \le\ C_\star\,\big(1+\mathbb{E}[Z^2]\big)\,\delta^{3/2},\qquad \delta^{3/2}=\delta\,\sqrt{\delta} .$$

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

$$\mathbb{E}\Big[|Z|\,\prod_{q=1}^{j}\big|\Delta M^{a_q}\big|\Big]\ \le\ C_\star\,\mathbb{E}\big[|Z|\big]\,\delta^2 .$$