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

Throughout, an observation clock index is a pair $d=(i,\upsilon)$, listed once and for all as a sequence of length $n=N\tilde{l}$, with channel $\upsilon$; $\tilde{Y}^d$ denotes the corresponding observation clock. For a record $r$ and an index $d=(i,\upsilon)$ we write $\tilde{A}^{r,d}=\tilde{A}^{r,i,\upsilon}$. We use freely the identity, valid for every $(r,\omega)\in G$ and $s\in[0,T]$, $\sum_{d}\tilde{\beta}\bigl(\sigma^{r,i}_s(\omega),\upsilon,\Sigma^r_s(\omega)\bigr)=N\,\tilde{b}^{\mathrm{tot}}\bigl(\Sigma^r_s(\omega)\bigr),\qquad \sum_{i}\tilde{\beta}\bigl(\sigma^{r,i}_s(\omega),\upsilon,\Sigma^r_s(\omega)\bigr)=N\,\tilde{b}^{\upsilon}\bigl(\Sigma^r_s(\omega)\bigr),$ obtained by grouping the agents by their state and using the aggregate observation drift, since $N\Sigma^{r,\sigma}_s$ agents occupy state $\sigma$; the same identities hold with all values replaced by left limits at a fixed time, the left-limit states forming a configuration with empirical vector $\Sigma^r_{t-}$.

Claim 1. By claim 4(a) and 4(c) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, $\rho(\mathbf{R})$ is the sum of the cell masses; the cell $C_\emptyset$ has mass $1$, and $C_{k,v}$ has mass $\lambda_k(D_k(T))=T^k/k!$ by claims 1 and 2 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions (restriction and transport preserve total mass) and The Ordered Time Simplex: Borel Measurability and Volume. Summing over the $\tilde{l}^k$ mark vectors for each $k$ and using that the sum in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions of nonnegative terms over a countable set is the limit of the partial sums along any exhausting sequence, $\rho(\mathbf{R})=\sum_{k\ge0}\tilde{l}^kT^k/k!=e^{\tilde{l}T}$, by the power series defining the real exponential function.

Claim 2. Fix a cell $C_{k,v}$. On $C_{k,v}\times\Omega$: each map $(r,\omega)\mapsto\Sigma^{r,\gamma}_{t_j(r)-}(\omega)=\frac{1}{N}\sum_i\eta^{r,i,\gamma}_{t_j(r)-}(\omega)$ is measurable by claim (b) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records; $\tilde{b}^{v_j}$ is a finite sum of products of coordinates with the rates $\tilde{\beta}(\sigma,v_j,\cdot)$, which are sequentially continuous by Observation-Rate Family, so the composition is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable. The map $((r,\omega),s)\mapsto\tilde{b}^{\mathrm{tot}}(\Sigma^r_s(\omega))$ is measurable by claim (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records and the same composition argument, and is bounded by $\tilde{l}\tilde{B}$; hence $(r,\omega)\mapsto\int_{[0,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^r_s(\omega))\,ds$ is measurable by the Tonelli theorem (sections and partial integrals of bounded jointly measurable maps), and composing with the continuous $\exp$ and multiplying finitely many real-valued measurable factors (Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable) shows that $f$ is measurable on $C_{k,v}\times\Omega$; likewise on $C_\emptyset\times\Omega$. Since the sets $C\times\Omega$ form a countable measurable partition of $\mathbf{R}\times\Omega$, $f$ is $\mathcal{R}\otimes\mathcal{T}$-measurable. The final assertion of claim 2 is the Tonelli theorem on the product of the finite measure spaces $(\Omega,\mathcal{T},P|_{\mathcal{T}})$ and $(\mathbf{R},\mathcal{R},\rho)$, the map $(r,\omega)\mapsto g(r)f(r,\omega)$ being $\mathcal{R}\otimes\mathcal{T}$-measurable.

Claim 3. Write $\Omega_0$ for the regular event of the solution and let $\Omega_1$ be the almost sure event on which every observation clock has finite, strictly increasing, unbounded jump times, provided by part (a) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law applied to each clock. For an observation clock index $d$ let $\xi^d_1,\xi^d_2,\dots$ be its interarrival times as in part (b) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law (set to $0$ off $\Omega_1$), so that on $\Omega_1$ the $m$-th jump time of $\tilde{Y}^d$ is $\xi^d_1+\dots+\xi^d_m$. Each $\xi^d_j$ is measurable with respect to the $\sigma$-algebra generated by the variables of $\tilde{Y}^d$ and the null events.

Step 1: the threshold vector. Fix a natural number $k\ge0$ and let $\Xi$ be the $\mathbb{R}^{n(k+1)}$-valued map with components $\xi^d_j$, $d$ an observation clock index and $1\le j\le k+1$. The components are independent: for events $\{\xi^d_j\in B^d_j\}$, grouping by clock, the intersections over $j$ are measurable with respect to the independent $\sigma$-algebras generated by the several clocks (by condition 4 of N-Agent Driving System and the following grouping argument: finitely many independent $\sigma$-algebras, partitioned into disjoint blocks, generate independent block $\sigma$-algebras — the finite intersections of members of a block form a $\pi$-system generating it, the product rule holds on these $\pi$-systems by the assumed independence, and it extends to the generated $\sigma$-algebras one block at a time by Dynkin's lemma, via claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law applied to the two finite measures obtained by fixing the other blocks' events; null completions change no probabilities), so the probability of the total intersection factors over clocks, and within each clock over $j$ by part (b) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law. By claim 1 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables, the distribution $\mu_\Xi$ of $\Xi$ is the product of the marginal distributions, each of which is the exponential law with density $h(u)=e^{-u}\mathbf{1}_{(0,\infty)}(u)$ with respect to Lebesgue measure, by claim 3 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law and part (b) of Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law. Moreover $\mu_\Xi$ is the measure with density $e(u)=\prod_{d,j}h(u_{d,j})$ with respect to $\lambda_{n(k+1)}$: both are probability measures on $\mathcal{B}_{n(k+1)}$ agreeing on the Borel rectangles, the product of the density measures having the rectangle values $\prod\int_{A_{d,j}}h\,d\lambda$ by claim 3 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and induction, so they coincide by claims 1 and 2 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law and the generation of $\mathcal{B}_{n(k+1)}$ by rectangles.

Step 2: the observation-event recursion. Fix also a mark vector $v\in V^k$. For $(\omega,u)\in\Omega\times\mathbb{R}^{n(k+1)}$ set $F(\omega,u)=0$ unless all $u_{d,j}>0$; when all $u_{d,j}>0$, define recursively: $R_0=r_\emptyset$, $T_0=0$, fired counts $q^d_0=0$; given stage $j$ with record-so-far $R_j$ (a record with $j$ events), time $T_j$, and counts $q^d_j$ with $\sum_d q^d_j=j$: for each $d$ let $\Theta^d_j=u_{d,1}+\dots+u_{d,q^d_j+1}$ and let $\chi^d_j$ be the least $s\in[T_j,T]$ with $\tilde{A}^{R_j,d}_s(\omega)\ge\Theta^d_j$ if one exists ($s\mapsto\tilde{A}^{R_j,d}_s$ is continuous and nondecreasing, by claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records and claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times), and $\chi^d_j=+\infty$ otherwise. If all $\chi^d_j=+\infty$, the recursion stops with $j$ events; otherwise the next event is at $T_{j+1}=\min_d\chi^d_j$, fired by the first minimizing clock $d_{j+1}$ in the listing, with channel $w_{j+1}$; put $R_{j+1}=(j+1,(T_1,\dots,T_{j+1}),(w_1,\dots,w_{j+1}))$ (a record when the times are strictly increasing; when $T_{j+1}=T_j$, stop and declare the outcome degenerate) and increment $q^{d_{j+1}}$. The recursion is run until it stops or until $j=k+1$. Let $F(\omega,u)=g(R_k)$ if the recursion stops after exactly $k$ nondegenerate events with mark vector $(w_1,\dots,w_k)=v$, and $F(\omega,u)=0$ otherwise. Measurability of $F$ with respect to $\mathcal{T}'\otimes\mathcal{B}_{n(k+1)}$, where $\mathcal{T}'$ is $\mathcal{T}$ enlarged by the null events, holds by induction on the stages: assuming the stage data $(T_j,q^d_j)$ and the map $(\omega,u)\mapsto R_j$ measurable into $(\mathbf{R},\mathcal{R})$ (cellwise, a vector of measurable times with fixed marks), the pairing $(\omega,u)\mapsto((R_j,s),\omega)$ composed with the jointly measurable field of claim (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records makes $((\omega,u),s)\mapsto\tilde{A}^{R_j,d}_s(\omega)$ measurable; the crossing characterization $\{\chi^d_j\le s\}=\{\tilde{A}^{R_j,d}_s\ge\Theta^d_j\}\cap\{T_j\le s\}$, valid by continuity, finite minima, and finitely many comparisons then make the next stage data measurable, and there are finitely many stages; $g$ enters through its cell restriction, a Borel function on $D_k(T)$ under the transport.

Step 3: identification and freezing. On $\Omega_0\cap\Omega_1\cap\Omega_G$, with $\Omega_G$ the full-probability event of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records with $\mathbf{R}\times\Omega_G\subseteq G$, the record satisfies: $W\in C_{k,v}$ if and only if the recursion at $(\omega,\Xi(\omega))$ stops after exactly $k$ nondegenerate events with marks $v$, and in that case the event times of $W$ are $(T_1,\dots,T_k)$; hence $\mathbf{1}_{C_{k,v}}(W)\,g(W)=F(\cdot,\Xi(\cdot))$ there. Indeed, by claim (f) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records the solution's consumed observation clock times equal $\tilde{A}^{W,d}$, and by claim (e) (causality) $\tilde{A}^{W,d}_s=\tilde{A}^{R_j,d}_s$ for $s$ before the $(j+1)$-th observation event time, and at that time as well by continuity of the consumed times (claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records and claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times), where $R_j$ is the record of the first $j$ events of $W$; the observation counter of clock $d$ is $\tilde{Y}^d$ evaluated along a continuous nondecreasing consumed time, so on $\Omega_1$ it jumps exactly when the consumed time crosses the next partial sum of the interarrivals; the observation events of the solution are therefore exactly the successive crossings of the recursion, the crossing clock at each event being unique on $\Omega_0$ (condition 3 of Solution of the Controlled N-Agent Dynamics: the observation total jumps by exactly one), so no degenerate outcome occurs and the marks agree. By condition 3 of Solution of the Controlled N-Agent Dynamics, the observation total coincides with the restriction of a counting path, so $K_T$ is finite on $\Omega_0$ and the recursion stops.

Let $Z$ be $\mathcal{T}$-measurable and nonnegative. The map $\Psi(\omega,u)=Z(\omega)F(\omega,u)$ is $\mathcal{T}'\otimes\mathcal{B}_{n(k+1)}$-measurable, and $\sigma(\Xi)$ is independent of $\mathcal{T}'$ (the interarrivals are measurable over the observation clocks and null events; independence of $\mathcal{T}'$ from the $\sigma$-algebra of all observation clocks follows from condition 4 of N-Agent Driving System by the grouping argument of Step 1). Since $\mathbf{1}_{C_{k,v}}(W)g(W)Z$ and $\Psi(\cdot,\Xi(\cdot))$ agree off the null event $\Omega\setminus(\Omega_0\cap\Omega_1\cap\Omega_G)$ and integrals of nonnegative measurable maps agreeing almost surely coincide, Independence Fubini: Integration in an Independent Random Vector Given a Sub-Sigma-Algebra, together with claim 3 of Image Measures, Measures with Densities, and Change of Variables to write the $\mu_\Xi$-integral as the $e\,\lambda_{n(k+1)}$-integral, gives $\mathbb{E}\bigl[Z\,g(W)\,\mathbf{1}_{C_{k,v}}(W)\bigr]=\mathbb{E}\Bigl[Z(\cdot)\int_{\mathbb{R}^{n(k+1)}}F(\cdot,u)\,e(u)\,d\lambda_{n(k+1)}(u)\Bigr].$

Step 4: evaluation of the threshold integral. Fix $\omega\in\Omega_G$ and abbreviate $\tilde{A}^{r,d}=\tilde{A}^{r,d}(\omega)$, $\Sigma^r=\Sigma^r(\omega)$. We claim $\int_{\mathbb{R}^{n(k+1)}}F(\omega,u)\,e(u)\,d\lambda_{n(k+1)}(u)=\int_{\mathbb{R}^k}\mathbf{1}_{D_k(T)}(t)\,g\bigl((k,t,v)\bigr)\,f\bigl((k,t,v),\omega\bigr)\,d\lambda_k(t),$ for every $\omega\in\Omega_G$; since $P(\Omega_G)=1$, this suffices for the expectations of Step 3, and combined with the cellwise decomposition below it proves claim 3. Decompose $F=\sum F_{\mathbf{d}}$ over the assignments $\mathbf{d}=(d_1,\dots,d_k)$ of firing clocks with channels $(v_1,\dots,v_k)$, the summands corresponding to disjoint events. Fix $\mathbf{d}$, and for clock $d$ let $p_d$ be the number of firings assigned to it. On the event described by $F_{\mathbf{d}}$, the coordinates $u_{d,j}$ with $j\ge p_d+2$ are unconstrained and integrate to $1$ against their density. By the Tonelli theorem and claim 3 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, integrate the remaining coordinates iteratedly: first, for fixed firing coordinates, the survival coordinates $u_{d,p_d+1}$: given that the $k$ events occur at times $t_1<\dots<t_k$ with final record $r=(k,t,v)$, clock $d$ produces no further event on its remaining threshold precisely when $u_{d,p_d+1}>\tilde{A}^{r,d}_T-\ell_d$, where $\ell_d$ is the consumed level of $d$ at its last assigned firing ($\ell_d=0$ if $p_d=0$), namely the sum of its firing coordinates, by the crossing identity $A(\kappa(w))=w$ of claim 3 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times applied at each firing; the integral over $u_{d,p_d+1}$ contributes $e^{-(\tilde{A}^{r,d}_T-\ell_d)}$, and here causality (claim (e) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records) identifies all record-so-far consumed times with those of the final record $r$ at the corresponding times. The product of these contributions with the firing densities $\prod_j e^{-w_j}$ telescopes: for each clock $d$, the exponents sum to $-\tilde{A}^{r,d}_T$, so the total exponential factor is $\exp(-\sum_d\tilde{A}^{r,d}_T)=\exp(-N\int_{[0,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^r_s)\,ds)$, by the agent-grouping identity and additivity of the integral.

Next integrate the firing coordinates $w_j=u_{d_j,\cdot}$ backwards, the last event first. For fixed $w_1,\dots,w_{j-1}$ (hence fixed $t_1<\dots<t_{j-1}$ and record prefix $R_{j-1}$), the $j$-th firing coordinate parameterizes the $j$-th event time as follows. Apply claim 4 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times on $[0,T]$ to the rate $s\mapsto\tilde{\beta}(\sigma^{R_{j-1},i_j}_s,\upsilon_j,\Sigma^{R_{j-1}}_s)\,\mathbf{1}\{s>t_{j-1}\}$, whose cumulative rate is $A_j(s)=\tilde{A}^{R_{j-1},d_j}_s-\tilde{A}^{R_{j-1},d_j}_{t_{j-1}}$, vanishing on $[0,t_{j-1}]$ so that crossings land in $(t_{j-1},T]$; the firing occurs where $A_j$ reaches $w_j$ shifted by the already-consumed excess $\tilde{A}^{R_{j-1},d_j}_{t_{j-1}}-\ell$, with $\ell$ the consumed level of $d_j$ at its previous firing, and this shift of the integration variable leaves the Lebesgue integral invariant by Translation Invariance of Lebesgue Measure and the Lebesgue Integral (applied to truncations and passed to the limit by the Monotone Convergence Theorem when the integrand is extended-valued); claim 4 then converts the $w_j$-integral into $\int_{[t_{j-1},T]}\Phi_j(s)\,\tilde{\beta}\bigl(\sigma^{R_{j-1},i_j}_s,\upsilon_j,\Sigma^{R_{j-1}}_s\bigr)\,ds,$ where $\Phi_j$ collects all factors depending on the $j$-th event time. Degenerate outcomes ($T_j=T_{j-1}$) and ties between clocks contribute nothing: for fixed values of the other coordinates each such outcome pins one firing coordinate to a single value, a $\lambda$-null set; so the assignment events may be treated as disjoint, with strict crossings, up to null sets. Since the reconstruction paths are piecewise constant, $\tilde{\beta}(\sigma^{R_{j-1},i_j}_s,\upsilon_j,\Sigma^{R_{j-1}}_s)$ coincides for all but finitely many $s$ with $\tilde{\beta}(\sigma^{R_j,i_j}_{s-},\upsilon_j,\Sigma^{R_j}_{s-})$ evaluated along the record extended by an event at $s$ (causality, and left limits at $s$ agree with the values at $s$ of the shorter record's path off the finitely many transition times), and finitely many points do not affect the integrals (claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval). Summing over the assignments $\mathbf{d}$ with the prescribed channels and using the agent-grouping identity at each event turns the product of the per-event rate factors into $\prod_{j=1}^k N\tilde{b}^{v_j}(\Sigma^r_{t_j-})$, by finitely many applications of linearity. Finally, the resulting iterated time-ordered integral over $0<t_1<\dots<t_k\le T$ of the nonnegative jointly measurable integrand $g((k,t,v))f((k,t,v),\omega)$ equals its $\lambda_k$-integral over $D_k(T)$, by the Tonelli theorem and induction on the coordinates exactly as in the volume computation of The Ordered Time Simplex: Borel Measurability and Volume. This proves the displayed identity; the case $k=0$ is the survival computation alone, giving $g(r_\emptyset)\exp(-N\int_{[0,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^{r_\emptyset}_s)ds)=g(r_\emptyset)f(r_\emptyset,\omega)\,\rho(C_\emptyset)$.

Conclusion of claim 3. Decompose $g=\sum_{C}g\,\mathbf{1}_{C}$ over the cells; by the Monotone Convergence Theorem and additivity applied to the partial sums on both sides, it suffices to prove the identity of claim 3 with $g$ replaced by $g\mathbf{1}_{C}$ for each cell $C$; unwinding $\int_{C_{k,v}}g\,f\,d\rho$ into the $\lambda_k$-integral over $D_k(T)$ (claims 1, 2, and 4(c) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions and claim 2 of Image Measures, Measures with Densities, and Change of Variables), this is exactly what Steps 3 and 4 establish.

Claim 4. Taking $g=1$ in claim 3, $\mathbb{E}[Z]=\mathbb{E}[Z\,I]$ for every $\mathcal{T}$-measurable $Z\ge0$, where $I=\int_{\mathbf{R}}f(r,\cdot)\,\rho(dr)$ is $\mathcal{T}$-measurable by claim 2. With $Z=\mathbf{1}_{\{I<1\}}$ this gives $\mathbb{E}[\mathbf{1}_{\{I<1\}}(1-I)]=0$, and the nonnegative random variable $\mathbf{1}_{\{I<1\}}(1-I)$ has expectation $0$, hence vanishes almost surely (if $P(I<1-1/n)>0$ for some $n$ the expectation would exceed $P(I<1-1/n)/n>0$), so $I\ge1$ almost surely; with $Z=\mathbf{1}_{\{I>1\}}$, likewise $I\le1$ almost surely, using on $\{I>1\}$ the truncations $\min(I,n)-1$ and the Monotone Convergence Theorem. Hence $I=1$ almost surely.

Claim 5. The map $p(r)=\mathbb{E}[f(r,\cdot)]$ is $\mathcal{R}$-measurable by the Tonelli theorem and finite, $f(r,\cdot)\le(N\tilde{B})^k$ on the cell with $k$ events. For $A\in\mathcal{R}$, claim 3 with $Z=1$ and $g=\mathbf{1}_A$ gives $P(W\in A)=\mathbb{E}\Bigl[\int_{\mathbf{R}}\mathbf{1}_A(r)f(r,\cdot)\,d\rho(r)\Bigr]=\int_{\mathbf{R}}\mathbf{1}_A\,p\,d\rho,$ by Tonelli; by claim 3 of Image Measures, Measures with Densities, and Change of Variables this identifies the image measure of $P$ under $W$ as the measure with density $p$ with respect to $\rho$.