Conditional Density of the Observation Record Given the Initial States and Transition Clocks

Adopt the setting of the controlled $N$-agent dynamics with $N\ge1$ agents, $l\ge2$ states, and $\tilde{l}\ge1$ observation channels: a transition-rate family $\beta$, 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 $h$, and a solution on $[0,T]$ with observation record $W$, which is measurable from $(\Omega,\mathcal{F})$ to the observation record space $(\mathbf{R},\mathcal{R})=(\mathbf{R}(T,\tilde{l}),\mathcal{R}(T,\tilde{l}))$ by claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records. Let $\rho$ be the reference measure of the observation record space, let $\mathcal{T}$ be the $\sigma$-algebra generated by the initial states and the transition-clock variables as in Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, and fix reconstruction data $(\eta^{r,i,\gamma},\sigma^{r,i},\tilde{A}^{r,i,\upsilon},G)$ with reconstructed empirical state measures $\Sigma^r$; the claims below hold for every admissible choice of such data.

Let $\tilde{b}$ be the aggregate observation drift of $\tilde{\beta}$ and define the total observation rate $\tilde{b}^{\mathrm{tot}}:\Delta^l\to\mathbb{R}$ on the probability simplex by $\tilde{b}^{\mathrm{tot}}(\Sigma)=\sum_{\upsilon=1}^{\tilde{l}}\tilde{b}^\upsilon(\Sigma)$, so that $0\le\tilde{b}^{\mathrm{tot}}\le\tilde{l}\tilde{B}$. The record density kernel is the function $f:\mathbf{R}\times\Omega\to[0,\infty)$ defined by $f(r,\omega)=0$ for $(r,\omega)\notin G$ and, for $(r,\omega)\in G$ with $r=(k,t,v)$ (so $k=0$ for $r=r_\emptyset$), $f(r,\omega)=\Bigl(\prod_{j=1}^{k}N\,\tilde{b}^{v_j}\bigl(\Sigma^r_{t_j-}(\omega)\bigr)\Bigr)\exp\Bigl(-N\int_{[0,T]}\tilde{b}^{\mathrm{tot}}\bigl(\Sigma^r_s(\omega)\bigr)\,ds\Bigr),$ with the empty product equal to $1$; here $\Sigma^r_{t-}$ denotes the left limit at $t$ of the path $s\mapsto\Sigma^r_s(\omega)$, which exists on the good set by claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records; $\exp$ is the real exponential function; the integrand is bounded and measurable on $[0,T]$ with the trace Borel $\sigma$-algebra, so the Lebesgue integral is finite; and on the cell with $k$ events, $0\le f\le(N\tilde{B})^k$.

1. (Reference measure) $\rho$ is a finite measure, with $\rho(\mathbf{R})=e^{\tilde{l}T}$.

2. (Kernel measurability) $f$ is measurable with respect to the product $\sigma$-algebra $\mathcal{R}\otimes\mathcal{T}$, by claims (a) and (b) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records; and for every $\mathcal{R}$-measurable $g:\mathbf{R}\to[0,\infty]$ the map $\omega\mapsto\int_{\mathbf{R}}g(r)\,f(r,\omega)\,\rho(dr)$ is $\mathcal{T}$-measurable, by the Tonelli theorem, both factors being finite.

3. (Conditional density) For every $\mathcal{T}$-measurable $Z:\Omega\to[0,\infty]$ and every $\mathcal{R}$-measurable $g:\mathbf{R}\to[0,\infty]$, $\mathbb{E}\bigl[Z\,g(W)\bigr]=\mathbb{E}\Bigl[Z\int_{\mathbf{R}}g(r)\,f(r,\cdot)\,\rho(dr)\Bigr]\qquad\text{in }[0,\infty],$ with expectations of $[0,\infty]$-valued measurable maps understood as their integrals with respect to $P$.

4. (Normalization) In particular, taking $g=1$ in claim 3: almost surely, $\int_{\mathbf{R}}f(r,\cdot)\,\rho(dr)=1$.

5. (Marginal density) The image measure of $P$ under $W$ is the measure with density $r\mapsto\mathbb{E}[f(r,\cdot)]$ with respect to $\rho$, this density being finite everywhere since $0\le f\le(N\tilde{B})^k$ on the cell with $k$ events.