One-Agent-Move Ratio of the Record Density Kernel

Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled $N$-agent dynamics with a transition-rate family $\beta$ with control set $\mathcal{A}$, an observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$ with initial states $\varsigma^i_0$ and clocks $Y^{i,\sigma\gamma}$, $\tilde{Y}^{i,\upsilon}$, an $\mathcal{A}$-valued observation-driven control policy $h$, and a solution on $[0,T]$; the observation record space $(\mathbf{R},\mathcal{R},\rho)$; the $\sigma$-algebra $\mathcal{T}$; fixed reconstruction data $(\eta^{r,i,\gamma},\sigma^{r,i},\tilde{A}^{r,i,\upsilon},G)$ with reconstructed empirical state measures $\Sigma^r$; the aggregate observation drift $\tilde{b}$ with total observation rate $\tilde{b}^{\mathrm{tot}}$; and the record density kernel $f$.

Fix an agent index $i_0\in\{1,\dots,N\}$ and states $\sigma^*,\gamma^*\in\{1,\dots,l\}$ with $\sigma^*\neq\gamma^*$, and define the moved initial states $\varsigma'^i_0$ by: $\varsigma'^i_0=\varsigma^i_0$ for $i\neq i_0$; $\varsigma'^{i_0}_0=\gamma^*$ on the event where $\varsigma^{i_0}_0=\sigma^*$, and $\varsigma'^{i_0}_0=\varsigma^{i_0}_0$ elsewhere.

1. (Moved driving system) The probability space $(\Omega,\mathcal{F},P)$ equipped with the moved initial states $\varsigma'^1_0,\dots,\varsigma'^N_0$ and the unchanged transition and observation clocks is again an $N$-agent driving system, called the moved system; a solution of the controlled $N$-agent dynamics on $[0,T]$ for the same policy $h$ on the moved system exists by Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics; and the $\sigma$-algebra $\mathcal{T}'$ generated by the moved initial states and the transition-clock variables --- the $\sigma$-algebra playing the role of $\mathcal{T}$ in Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records and Conditional Density of the Observation Record Given the Initial States and Transition Clocks when they are instantiated on the moved system --- satisfies $\mathcal{T}'\subseteq\mathcal{T}$. Fix reconstruction data $(\eta'^{r,i,\gamma},\sigma'^{r,i},\tilde{A}'^{r,i,\upsilon},G')$ for the moved system and the policy $h$, with reconstructed empirical state measures $\Sigma'^r$, and let $f':\mathbf{R}\times\Omega\to[0,\infty)$ be defined exactly as the record density kernel, with the moved reconstruction data in place of the original data; all claims of Conditional Density of the Observation Record Given the Initial States and Transition Clocks hold for the moved system, its setting being instantiated by the moved driving system, the policy $h$, and a moved solution. The claims below hold for every admissible choice of the two reconstruction data sets.

2. (Positivity) For every $(r,\omega)\in G$ with $r=(k,t,v)$: $f(r,\omega)>0$ if and only if $\tilde{b}^{v_j}(\Sigma^r_{t_j-}(\omega))>0$ for every $j\in\{1,\dots,k\}$; and likewise for $f'$ on $G'$ with $\Sigma'^r$ in place of $\Sigma^r$.

3. (One-agent-move likelihood ratio) Let $\Omega_G$ and $\Omega'_G$ be events of probability one with $\mathbf{R}\times\Omega_G\subseteq G$ and $\mathbf{R}\times\Omega'_G\subseteq G'$, as provided by Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records. For every $\omega\in\Omega_G\cap\Omega'_G$ and every $r=(k,t,v)\in\mathbf{R}$ with $f(r,\omega)>0$, $\frac{f'(r,\omega)}{f(r,\omega)}=\Biggl(\prod_{j=1}^{k}\frac{\tilde{b}^{v_j}\bigl(\Sigma'^r_{t_j-}(\omega)\bigr)}{\tilde{b}^{v_j}\bigl(\Sigma^r_{t_j-}(\omega)\bigr)}\Biggr)\exp\Bigl(-N\int_{[0,T]}\bigl(\tilde{b}^{\mathrm{tot}}\bigl(\Sigma'^r_u(\omega)\bigr)-\tilde{b}^{\mathrm{tot}}\bigl(\Sigma^r_u(\omega)\bigr)\bigr)\,du\Bigr),$ with the empty product equal to $1$ (the right side being $0$ exactly when some $\tilde{b}^{v_j}(\Sigma'^r_{t_j-}(\omega))=0$, no positivity of $f'$ being assumed), $\exp$ the real exponential function, and the integral over the compact interval the Lebesgue integral of a bounded measurable integrand, which exists since both terms are bounded by $\tilde{l}\tilde{B}$.

4. (Measurability of the ratio field) The map on $\mathbf{R}\times\Omega$ that equals $f'(r,\omega)/f(r,\omega)$ where $f(r,\omega)>0$ and equals $0$ where $f(r,\omega)=0$ is measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ (product $\sigma$-algebra).