Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics

Let $N$, $l$, $\tilde{l}$, $m$ be natural numbers with $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, let $\beta$ be a transition-rate family on $l$ states with control dimension $m$ and rate bound $B$, let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, and let $T>0$ be a real number.

(i) (Driving systems exist.) For every family $(p_x)$ of nonnegative real numbers indexed by $x\in\{1,\dots,l\}^N$ with $\sum_x p_x=1$, there exists an $N$-agent driving system whose initial states satisfy $P(\varsigma^1_0=x_1,\dots,\varsigma^N_0=x_N)=p_x$ for every $x=(x_1,\dots,x_N)$.

(ii) (Existence and uniqueness of solutions.) For every $N$-agent driving system $(\Omega,\mathcal{F},P)$ and every observation-driven control policy $h$ with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, there exists a solution of the controlled $N$-agent dynamics on $[0,T]$; and any two solutions $(\sigma^i,\Upsilon^\upsilon,\alpha)$ and $(\hat{\sigma}^i,\hat{\Upsilon}^\upsilon,\hat{\alpha})$ are indistinguishable: almost surely, for all $t\in[0,T]$, $\sigma^i_t=\hat{\sigma}^i_t$ for all $i$, $\Upsilon^\upsilon_t=\hat{\Upsilon}^\upsilon_t$ for all $\upsilon$, and $\alpha_t=\hat{\alpha}_t$.

(iii) (Event count bound.) For any solution, almost surely, for all $t\in[0,T]$,

where the sums run over $i\in\{1,\dots,N\}$, ordered pairs $(\sigma,\gamma)$ with $\sigma\neq\gamma$, and $\upsilon\in\{1,\dots,\tilde{l}\}$. In particular, since each variable on the right has the Poisson distribution and finite second moment, every counter $N^{i,\sigma\gamma}_t$ and $\tilde{N}^{i,\upsilon}_t$ is square-integrable.

(iv) (Adaptedness.) For any solution, the processes $\sigma^i$, $\eta^{i,\gamma}$, $\Sigma^\gamma$, $N^{i,\sigma\gamma}$, $\tilde{N}^{i,\upsilon}$, $A^{i,\sigma\gamma}$, $\tilde{A}^{i,\upsilon}$, and $\Upsilon^\upsilon$ are adapted to the system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ of the solution; for every $t\in[0,T]$, the observation-event count $K_t$, the observation event times $\tau_j$ with $j\le K_t$, and their channels $\upsilon_j$ with $j\le K_t$ are measurable with respect to the observation filtration $\mathcal{G}_t$ of the solution; and each component $\alpha^j_t$ of the control is $\mathcal{G}_t$-measurable.

(v) (Cost well-definedness.) Let $(L,G)$ be population cost data on $l$ states with control dimension $m$. For any solution, almost surely the path $t\mapsto L(\Sigma_t,\alpha_t)$ is measurable on $[0,T]$ with the trace Borel $\sigma$-algebra and is bounded below by $-C_L$, so that the Lebesgue integral $\int_{[0,T]}L(\Sigma_t,\alpha_t)\,dt$ exists in $(-\infty,+\infty]$. Moreover the map

is measurable as an extended-real-valued map (the sets on which it is at most $a$ are events for every real $a$), it is bounded below by the constant $-C_LT-C_G$, and therefore its expectation is well defined in $(-\infty,+\infty]$ as the limit of the expectations of its truncations at level $n$ as $n\to\infty$.

(vi) (Clock-reading bound.) For any solution, fix $r\in[0,T]$ and nonnegative real numbers $c_{i,\sigma\gamma}$ (one for each transition clock) and $c_{i,\upsilon}$ (one for each observation clock), and let $C$ be the event that $A^{i,\sigma\gamma}_r\le c_{i,\sigma\gamma}$ and $\tilde{A}^{i,\upsilon}_r\le c_{i,\upsilon}$ for all indices. Let $\mathcal{H}$ denote the $\sigma$-algebra generated by the initial states $\varsigma^1_0,\dots,\varsigma^N_0$ together with the clock variables $Y^{i,\sigma\gamma}_u$ for $0\le u\le c_{i,\sigma\gamma}$ and $\tilde{Y}^{i,\upsilon}_u$ for $0\le u\le c_{i,\upsilon}$ (all indices). Then the event $C$ agrees up to an event of probability zero with an event of $\mathcal{H}$, and for every event $F\in\mathcal{F}^{\mathrm{sys}}_r$ there is an event $H\in\mathcal{H}$ such that the symmetric difference of $F\cap C$ and $H\cap C$ is an event of probability zero.