Moment Bounds for the Aggregate Compensated Counters of the Controlled N-Agent Dynamics

Adopt the setting of the compensated counters of the controlled $N$-agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$: a transition-rate family $\beta$ with rate bound $B$, an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$, a solution on $[0,T]$ with regular event $\Omega_0$, counters $N^{i,\sigma\gamma}_t$, consumed clock times $\mathcal{T}^{i,\sigma\gamma}_t$, and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and the compensated counters $M^{i,\sigma\gamma}_t=N^{i,\sigma\gamma}_t-\mathcal{T}^{i,\sigma\gamma}_t$ of that lemma. Write $\mathbb{E}$ for the expectation. For each ordered pair $(\sigma,\gamma)$ of distinct states define the aggregate counter, the aggregate consumed clock time, and the aggregate compensated counter

$$\mathcal{N}^{\sigma\gamma}_t=\sum_{i=1}^{N}N^{i,\sigma\gamma}_t,\qquad \mathfrak{A}^{\sigma\gamma}_t=\sum_{i=1}^{N}\mathcal{T}^{i,\sigma\gamma}_t,\qquad \mathfrak{M}^{\sigma\gamma}_t=\mathcal{N}^{\sigma\gamma}_t-\mathfrak{A}^{\sigma\gamma}_t=\sum_{i=1}^{N}M^{i,\sigma\gamma}_t\qquad(t\in[0,T]);$$

the fraktur $\mathfrak{A}^{\sigma\gamma}$, a sum of consumed clock times, is unrelated to the calligraphic control-energy symbol $\mathcal{A}$ used in companion results of this series. Then, for every ordered pair $(\sigma,\gamma)$ of distinct states:

(a) (Martingale, counting-path, and compensator properties.) $(\mathfrak{M}^{\sigma\gamma}_t)_{t\in[0,T]}$ is a square-integrable martingale with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ with $\mathfrak{M}^{\sigma\gamma}_0=0$; almost surely, $t\mapsto\mathcal{N}^{\sigma\gamma}_t$ agrees on $[0,T]$ with a counting path; and at every $\omega\in\Omega$ and all $0\le r\le t\le T$,

$$0\ \le\ \mathfrak{A}^{\sigma\gamma}_t-\mathfrak{A}^{\sigma\gamma}_r\ \le\ N\,B\,(t-r).$$

(b) (Second and fourth moments.) For every $t\in[0,T]$, the powers $(\mathfrak{M}^{\sigma\gamma}_t)^2$ and $(\mathfrak{M}^{\sigma\gamma}_t)^4$ are integrable, and

$$\mathbb{E}\big[(\mathfrak{M}^{\sigma\gamma}_t)^2\big]=\mathbb{E}\big[\mathfrak{A}^{\sigma\gamma}_t\big]\le N\,B\,t,\qquad\qquad \mathbb{E}\big[(\mathfrak{M}^{\sigma\gamma}_t)^4\big]\le 6\,\big(N\,B\,t+(N\,B\,t)^2\big).$$

(c) (Aggregate state martingale.) Let $M_t=(M^\gamma_t)_{\gamma\in\{1,\dots,l\}}$ be as in the martingale decomposition (the superscript shapes distinguish the per-clock compensated counters $M^{i,\sigma\gamma}$, the aggregate compensated counters $\mathfrak{M}^{\sigma\gamma}$, and the state martingales $M^\gamma$), and write $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin) and $\sqrt{N}$ for the nonnegative square root. Then, almost surely, for every $\gamma\in\{1,\dots,l\}$ and every $t\in[0,T]$,

$$N\,M^\gamma_t=\sum_{\sigma:\sigma\neq\gamma}\Big(\mathfrak{M}^{\sigma\gamma}_t-\mathfrak{M}^{\gamma\sigma}_t\Big),$$

the sum running over $\sigma\in\{1,\dots,l\}$ with $\sigma\neq\gamma$; and consequently, for every $t\in[0,T]$,

$$\mathbb{E}\big[|\sqrt{N}\,M_t|^4\big]\ \le\ 6\,l^2\,\big(2(l-1)\big)^4\,\Big(\frac{B\,t}{N}+(B\,t)^2\Big)\ \le\ 6\,l^2\,\big(2(l-1)\big)^4\,\big(B\,t+(B\,t)^2\big).$$