Proof of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics

Throughout, fix $r\in\mathbf{R}$ and write $\mathfrak{S}$ for the solution furnished by clause (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records: its state processes are the $\sigma^{r,i}$, its control process is $\alpha_s=a^r(s)$, its regular event is $\Omega^r$ with $P(\Omega^r)=1$, and $\{(r,\omega):\omega\in\Omega^r\}\subseteq G$. Its clocks are the transition and observation clocks of the driving system. We write $X^i_t=\sigma^{r,i}_t$ for the $i$-th coordinate of $X_t$. Measurability of real-valued maps on $[0,T]$ or on $\Omega$ is always with respect to $\mathcal{B}_{[0,T]}$ or $\mathcal{F}$ and the Borel $\sigma$-algebra of the real line, and Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions is used for constants, indicators, sums and products of measurable maps.

Step 0 ($\Sigma(x)$ lies in the lattice). For $x\in E$ the counts $\#\{i:x^i=\gamma\}$, $\gamma\in\{1,\dots,l\}$, are nonnegative integers with sum $N$, each index $i$ having exactly one value $x^i$. Hence the coordinates of $\Sigma(x)$ are nonnegative with sum $1$, so $\Sigma(x)$ lies in the probability simplex $\Delta^l$, and $N\Sigma(x)^\gamma$ is $0$ or a natural number; thus $\Sigma(x)\in\mathbb{G}_N$ by the definition of the aggregate lattice. In particular $\mathbb{G}_N$ is nonempty, and it is finite, being contained in the set of points of $\mathbb{R}^l$ all of whose coordinates lie in $\{0,\frac1N,\dots,1\}$, which has $(N+1)^l$ elements; $E$ has $l^N\ge1$ elements and $\mathsf{A}$ has $Nl(l-1)\ge1$ elements, so the finiteness and nonemptiness hypotheses of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property and Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set hold.

Step 1 (Identities). By the derived notation of Solution of the Controlled N-Agent Dynamics, the occupation indicators of $\mathfrak{S}$ are $\eta^{i,\gamma}_t=\mathbf{1}\{X^i_t=\gamma\}$ and its empirical state measure is the point with coordinates $\frac1N\sum_i\mathbf{1}\{X^i_t=\gamma\}=\Sigma(X_t)^\gamma$, that is, $\hat{\Sigma}^r_t$, at every $\omega\in\Omega$. Now let $\omega\in\Omega^r$. Then $(r,\omega)\in G$, so clause (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records gives, for every $i$ and $t$, exactly one $\gamma$ with $\eta^{r,i,\gamma}_t(\omega)=1$, the others being $0$, and $\sigma^{r,i}_t(\omega)$ is that $\gamma$; hence $\eta^{r,i,\gamma}_t(\omega)=\mathbf{1}\{\sigma^{r,i}_t(\omega)=\gamma\}$ for all $\gamma$, and therefore $\Sigma^r_t(\omega)=\hat{\Sigma}^r_t(\omega)$. This proves the first assertion of claim 1. Consequently, by condition 2 of Solution of the Controlled N-Agent Dynamics for $\mathfrak{S}$ (regular event $\Omega^r$, control $a^r$), for every $\omega\in\Omega$, every $e=(i,(\sigma,\gamma))\in\mathsf{A}$ and every $t\in[0,T]$,

Step 2 (D1). By condition 2 of N-Agent Driving System, each $Y^{i,\sigma\gamma}$ is a stochastic process, that is, a family of random variables indexed by $[0,\infty)$ (the earlier version of that definition invoked in N-Agent Driving System defines the term identically), every path of which is a counting path.

Step 3 (D2). Fix $e=(i,(\sigma,\gamma))$ and $x\in E$. By clause (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, $a^r(u)\in\mathcal{A}$ for every $u$, and $\Sigma(x)\in\Delta^l$ by Step 0, so $0\le g^{e}(u,x)\le B$ by condition 1 of Transition-Rate Family. If $x^i\neq\sigma$ then $g^{e}(\cdot,x)$ is the constant $0$, which is measurable. If $x^i=\sigma$ then $g^{e}(u,x)=\beta(\sigma,\gamma,y,a^r(u))$ with $y=\Sigma(x)$; we show more generally that $u\mapsto\beta(\sigma,\gamma,y,a^r(u))$ is measurable for every $y\in\Delta^l$. The map $\alpha\mapsto\beta(\sigma,\gamma,y,\alpha)$ is sequentially continuous on $\mathcal{A}\subseteq\mathbb{R}^m$: if $\zeta_n\to\zeta$ in $\mathcal{A}$ in Euclidean distance, then the sequence $(y,\zeta_n)$ satisfies the hypothesis of condition 2 of Transition-Rate Family with constant first entries, whence $\beta(\sigma,\gamma,y,\zeta_n)\to\beta(\sigma,\gamma,y,\zeta)$. Each component of $a^r:[0,T]\to\mathcal{A}$ is measurable with respect to $\mathcal{B}_{[0,T]}$ by The Record-Frozen Control Path and Record-Frozen Policy. Hence $u\mapsto\beta(\sigma,\gamma,y,a^r(u))$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, applied on the measurable space $([0,T],\mathcal{B}_{[0,T]})$ with $d=m$, the subset $\mathcal{A}$ and the map $a^r$. Since $a^r$ takes values in $\mathcal{A}$ (clause (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records) and its components are $\mathcal{B}_{[0,T]}$-measurable (The Record-Frozen Control Path and Record-Frozen Policy), $a^r$ is a control path in the sense of that definition; this is the assertion of claim 1 concerning $a^r$.

Step 4 (D3). The system filtration $(\mathcal{F}^{\mathrm{sys},r}_t)_{t\in[0,T]}$ of $\mathfrak{S}$ is a filtration of sub-$\sigma$-algebras of $\mathcal{F}$ by clause (vii)(e) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, and $\Omega^r\in\mathcal{F}$ has probability one. For $t\in[0,T]$ and $x\in E$, $\{X_t=x\}=\bigcap_{i=1}^N\{\sigma^{r,i}_t=x^i\}$, and each $\{\sigma^{r,i}_t=x^i\}$ lies in $\mathcal{F}^{\mathrm{sys},r}_t$ because $\sigma^{r,i}$ is adapted to the system filtration by clause (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics; so $\{X_t=x\}\in\mathcal{F}^{\mathrm{sys},r}_t$.

For the product measurability, fix $x\in E$. By Step 1, for every $(u,\omega)\in[0,T]\times\Omega$,

since on $\Omega^r$ one has $\mathbf{1}\{X^i_u=x^i\}=\eta^{r,i,x^i}_u$ for every $i$, and both sides vanish off $\Omega^r$. The first factor is the indicator of the measurable rectangle $[0,T]\times\Omega^r$ and is therefore $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable. For the remaining factors we show that for all $i,\gamma$ the map $(u,\omega)\mapsto\eta^{r,i,\gamma}_u(\omega)$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{T}$-measurable; this suffices, because $\mathcal{T}\subseteq\mathcal{F}$ makes every measurable rectangle of $\mathcal{B}_{[0,T]}\otimes\mathcal{T}$ a measurable rectangle of $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$, so that $\mathcal{B}_{[0,T]}\otimes\mathcal{T}\subseteq\mathcal{B}_{[0,T]}\otimes\mathcal{F}$ by the definition of the product $\sigma$-algebra as a generated $\sigma$-algebra, and a finite product of measurable real-valued maps is measurable. Consider the map $\iota:[0,T]\times\Omega\to(\mathbf{R}\times[0,T])\times\Omega$, $\iota(u,\omega)=((r,u),\omega)$. The family of subsets $S$ of $(\mathbf{R}\times[0,T])\times\Omega$ with $\iota^{-1}(S)\in\mathcal{B}_{[0,T]}\otimes\mathcal{T}$ is a $\sigma$-algebra (preimages commute with complements and countable unions), so $\iota$ is measurable from $\mathcal{B}_{[0,T]}\otimes\mathcal{T}$ to $(\mathcal{R}\otimes\mathcal{B}_{[0,T]})\otimes\mathcal{T}$ as soon as $\iota^{-1}(A\times C)\in\mathcal{B}_{[0,T]}\otimes\mathcal{T}$ for every $A\in\mathcal{R}\otimes\mathcal{B}_{[0,T]}$ and $C\in\mathcal{T}$. Now $\iota^{-1}(A\times C)=A_r\times C$ with $A_r=\{u\in[0,T]:(r,u)\in A\}$. The family $\mathcal{S}$ of subsets $A$ of $\mathbf{R}\times[0,T]$ with $A_r\in\mathcal{B}_{[0,T]}$ is a $\sigma$-algebra, since $(\mathbf{R}\times[0,T]\setminus A)_r=[0,T]\setminus A_r$, $(\bigcup_nA_n)_r=\bigcup_n(A_n)_r$, and $\mathcal{B}_{[0,T]}$ is a $\sigma$-algebra by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; and $\mathcal{S}$ contains every measurable rectangle $A'\times B'$ with $A'\in\mathcal{R}$ and $B'\in\mathcal{B}_{[0,T]}$, whose $r$-section is $B'$ if $r\in A'$ and empty otherwise. Hence $\mathcal{S}\supseteq\mathcal{R}\otimes\mathcal{B}_{[0,T]}$, so $A_r\in\mathcal{B}_{[0,T]}$ and $A_r\times C$ is a measurable rectangle of $\mathcal{B}_{[0,T]}\otimes\mathcal{T}$. Thus $\iota$ is measurable, and $(u,\omega)\mapsto\eta^{r,i,\gamma}_u(\omega)$ is the composition of $\iota$ with the map $(r',u,\omega)\mapsto\eta^{r',i,\gamma}_u(\omega)$, which is $(\mathcal{R}\otimes\mathcal{B}_{[0,T]})\otimes\mathcal{T}$-measurable by clause (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records; a composition of measurable maps is measurable, the preimage of a set under the composition being the preimage under $\iota$ of its preimage under the outer map. The same argument with the map $\omega\mapsto((r,u),\omega)$ for fixed $u$ (preimages of rectangles $A\times C$ are $C$ or empty) shows that $\omega\mapsto\eta^{r,i,\gamma}_u(\omega)$ is $\mathcal{T}$-measurable for each fixed $u$, so that $\{\Sigma^r_u=y\}\in\mathcal{T}\subseteq\mathcal{F}$ for every $u\in[0,T]$ and $y\in\mathbb{R}^l$; this is used in Step 8.

Step 5 (D4). The solution $\mathfrak{S}$ satisfies conditions 1--4 of Solution of the Controlled N-Agent Dynamics with regular event $\Omega^r$, so clause (vii)(b) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics shows that each $\mathcal{T}^{i,\sigma\gamma}_t$ is $\mathcal{F}$-measurable with values in $[0,Bt]=[0,\Lambda t]$; and by (1), for $\omega\in\Omega^r$, $\mathcal{T}^{i,\sigma\gamma}_t(\omega)=\int_{[0,t]}g^{e}(u,X_u(\omega))\,du$ with $e=(i,(\sigma,\gamma))$, as (D4) requires. The counters of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property are $Y^{e}_{\mathcal{T}^{e}_t}=Y^{i,\sigma\gamma}_{\mathcal{T}^{i,\sigma\gamma}_t}=N^{i,\sigma\gamma}_t$, the transition counters of condition 3 of Solution of the Controlled N-Agent Dynamics.

Step 6 (H1). Fix $\omega\in\Omega^r$; conditions 1, 3 and 6 of Solution of the Controlled N-Agent Dynamics hold at $\omega$, and we suppress $\omega$.

Piecewise constancy. By condition 1, for each $i$ there are a count $K^{(i)}$ and times $0<t^{(i)}_1<\dots<t^{(i)}_{K^{(i)}}\le T$ such that $X^i$ is constant on $[0,t^{(i)}_1)$, on each $[t^{(i)}_j,t^{(i)}_{j+1})$, and on $[t^{(i)}_{K^{(i)}},T]$ (on all of $[0,T]$ if $K^{(i)}=0$). Let $0<t_1<\dots<t_K\le T$ enumerate the union of these finite sets of times ($K=0$ if the union is empty). Let $I$ be one of the intervals $[0,t_1)$, $[t_k,t_{k+1})$ ($1\le k<K$), $[t_K,T]$ (or $[0,T]$ if $K=0$), with left endpoint $\ell$. Fix $i$ and let $j$ be the largest index with $t^{(i)}_j\le\ell$, or $j=0$ if there is none. If $j<K^{(i)}$ then $t^{(i)}_{j+1}>\ell$, and since $t^{(i)}_{j+1}$ belongs to the union it is at least the next enumerated time after $\ell$, which is the right endpoint of $I$ when $I=[t_k,t_{k+1})$ or $[0,t_1)$, and does not exist when $I=[t_K,T]$ (all enumerated times being $\le t_K$), so that this case is impossible then. Hence $I$ is contained in the constancy interval of $X^i$ starting at $t^{(i)}_j$ (at $0$ if $j=0$), and $X^i$ is constant on $I$. As this holds for every $i$, $X$ is constant on $I$: the path of $X$ is piecewise constant and right-continuous with the times $t_1,\dots,t_K$, as (H1) requires, and $X_{t-}$ is defined for $t\in(0,T]$.

Jump rule. Let $t\in(0,T]$, and let $N^{e}_{t-}$ and the notation of (H1) be as in Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property, and write $\mathcal{N}_t=\sum_{e\in\mathsf{A}}N^{e}_t$ and $\mathcal{N}_{t-}$ for the grand total and its left limit, called $N_t$ and $N_{t-}$ there (the bare symbol $N$ denotes the number of agents here); put $\Delta N^{e}_t=N^{e}_t-N^{e}_{t-}$, a nonnegative integer. By condition 3 each path $s\mapsto N^{e}_s$ coincides on $[0,T]$ with a counting path, so it is nondecreasing and integer-valued; the set $\{N^{e}_s:0\le s<t\}$ is a set of integers bounded above by $N^{e}_t$, so its supremum $N^{e}_{t-}$ is attained at some $s_e<t$, and by monotonicity $N^{e}_s=N^{e}_{t-}$ for all $s\in[s_e,t)$. Choose $s'<t$ with $s'\ge s_e$ for all $e\in\mathsf{A}$ and such that $X_s=X_{t-}$ for $s\in(s',t)$ (possible by piecewise constancy). Fix $s\in(s',t)$. Condition 6 at the times $t$ and $s$, subtracted, gives for all $i$ and $\gamma$, using $\eta^{i,\gamma}=\mathbf{1}\{X^i=\gamma\}$ from Step 1,

If $\mathcal{N}_t=\mathcal{N}_{t-}$, then, as $\mathcal{N}_{t-}=\sum_eN^{e}_{t-}$, all $\Delta N^{e}_t$ vanish, so (2) gives $\mathbf{1}\{X^i_t=\gamma\}=\mathbf{1}\{X^i_{t-}=\gamma\}$ for all $i,\gamma$, that is, $X_t=X_{t-}$. If $\mathcal{N}_t=\mathcal{N}_{t-}+1$, let $e_0=(i_0,(\sigma_0,\gamma_0))$ be the unique label with $\Delta N^{e_0}_t=1$, all other $\Delta N^{e}_t$ being $0$. For $i\neq i_0$ the right side of (2) vanishes for every $\gamma$, so $X^i_t=X^i_{t-}$. For $i=i_0$ and $\gamma=\gamma_0$, the right side of (2) equals $\Delta N^{i_0,\sigma_0\gamma_0}_t=1$, no label of the form $(i_0,(\gamma_0,\gamma'))$ being $e_0$ since $\gamma_0\neq\sigma_0$; hence $\mathbf{1}\{X^{i_0}_t=\gamma_0\}=1+\mathbf{1}\{X^{i_0}_{t-}=\gamma_0\}\ge1$, forcing $X^{i_0}_t=\gamma_0$. Thus $X_t=\phi^{e_0}(X_{t-})$, as (H1) requires.

Step 7 (H2). Fix $s\in[0,T]$, the time called $r$ in (H2), and put $\varrho_s=B(T-s)$. The residual clocks of (H2) are $\hat{Y}^{e,s}_u=Y^{i,\sigma\gamma}_{\mathcal{T}^{i,\sigma\gamma}_s+u}-Y^{i,\sigma\gamma}_{\mathcal{T}^{i,\sigma\gamma}_s}$ for $e=(i,(\sigma,\gamma))$, which are the residual transition clocks $\hat{Y}^{i,\sigma\gamma}_u$ of Fresh-Start Property of the Controlled N-Agent Dynamics applied at time $s$ to the solution $\mathfrak{S}$; that lemma applies with the policy $\hat{h}^r$, which is $\mathcal{A}$-valued by clause (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, with the solution $\mathfrak{S}$ on the same driving system, and with its time parameter equal to $s$. By its clause (a), each residual clock is a homogeneous Poisson process with rate $1$: each $\hat{Y}^{e,s}_u$ is a random variable, and by conditions 2 and 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process (the earlier version of that definition, which is the one invoked in Fresh-Start Property of the Controlled N-Agent Dynamics, states conditions 1--3 and the notion of a homogeneous Poisson process with rate $1$ identically; the two versions differ only in the continuity requirement imposed on a general intensity function, which plays no role here), for all real $0=u_0<u_1<\dots<u_p\le\varrho_s$ the increments $\hat{Y}^{e,s}_{u_q}-\hat{Y}^{e,s}_{u_{q-1}}$, $1\le q\le p$, are independent, the $q$-th having the Poisson distribution with parameter $u_q-u_{q-1}$. For the independence requirement, clause (b) of Fresh-Start Property of the Controlled N-Agent Dynamics states that the family consisting of $\mathcal{F}^{\mathrm{sys},r}_s$, the $\sigma$-algebras $\sigma(\hat{Y}^{i,\sigma\gamma}_u:u\ge0)$ and the $\sigma$-algebras generated by the residual observation clocks is independent. The family required by (H2) consists of $\mathcal{F}^{\mathrm{sys},r}_s$ and the $\sigma$-algebras $\mathcal{G}_e=\sigma(\hat{Y}^{e,s}_u:0\le u\le\varrho_s)$, $e\in\mathsf{A}$; each $\mathcal{G}_e$ is contained in $\sigma(\hat{Y}^{i,\sigma\gamma}_u:u\ge0)$, its generators in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras being among those of the latter, so that the smaller generated $\sigma$-algebra is contained in the larger. Given finitely many distinct members of the (H2) family and events in them, these events lie in the corresponding distinct members of the fresh-start family, so the product identity of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras holds. Thus (H2) holds.

Step 8 (Claims 1 and 2). Steps 2--7 establish (D1)--(D4), (H1) and (H2), which proves claim 1, so conclusions (a) and (b) of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property hold: for every $\tilde{F}:E\to\mathbb{R}$, all $0\le s\le t\le T$ and $D\in\mathcal{F}^{\mathrm{sys},r}_s$,

the inner integral being defined everywhere and $\mathcal{F}$-measurable; and for $s\in[0,T)$ and $D\in\mathcal{F}^{\mathrm{sys},r}_s$ the family $\mu^{D}_u(x)=P(D\cap\{X_u=x\})$, $u\in[s,T]$, is a solution of the forward equation on $[s,T]$ for the rates $q^{E}_u(x,x')=\sum_{e:\phi^{e}(x)=x'}g^{e}(u,x)$ on $E$ (called $q$ in that lemma).

Generator identity. Let $F:\mathbb{G}_N\to\mathbb{R}$, with $F(y+\frac1Nv_c)$ read as $F(y)$ when $y+\frac1Nv_c\notin\mathbb{G}_N$. We claim that for all $u\in[0,T]$ and $x\in E$, writing $y=\Sigma(x)$,

For the first equality fix $c=(\sigma,\gamma)$ and consider the labels $e=(i,c)$. If $x^i\neq\sigma$ then $g^{e}(u,x)=0$. If $x^i=\sigma$, then $\phi^{e}(x)$ differs from $x$ only in coordinate $i$, which changes from $\sigma$ to $\gamma$, so the count of $\sigma$ decreases by one and that of $\gamma$ increases by one: $\Sigma(\phi^{e}(x))=y+\frac1N(\delta_\gamma-\delta_\sigma)=y+\frac1Nv_c$, a point of $\mathbb{G}_N$ by Step 0, and $g^{e}(u,x)=\beta(\sigma,\gamma,y,a^r(u))$. There are $Ny^{\sigma}$ indices $i$ with $x^i=\sigma$, so the labels $(i,c)$ contribute $Ny^{\sigma}\beta(\sigma,\gamma,y,a^r(u))(F(y+\frac1Nv_c)-F(y))$ to the left side, which is also the $c$-term of the middle expression; when $y^{\sigma}=0$ both are $0$ whatever the reading of $F(y+\frac1Nv_c)$. For the second equality: if $y^{\sigma}\ge\frac1N$ then $y+\frac1Nv_c$ has nonnegative coordinates $y^{\sigma}-\frac1N$, $y^{\gamma}+\frac1N$ and $y^{\gamma'}$ ($\gamma'\notin\{\sigma,\gamma\}$), all multiples of $\frac1N$ with sum $1$, so it lies in $\mathbb{G}_N$, whereas if $y^{\sigma}=0$ its $\sigma$-coordinate is negative and it does not; distinct labels give distinct points $y+\frac1Nv_c$, as $v_c$ determines $c$. Hence, by the definition of $q^r$, the right side of (4) is the sum over the labels $c$ with $y^{\sigma}\ge\frac1N$ of $Ny^{\sigma}\beta(\sigma,\gamma,y,a^r(u))(F(y+\frac1Nv_c)-F(y))$, and the remaining terms of the middle expression vanish.

The displayed identity of claim 2. Apply (3) with $\tilde{F}=F\circ\Sigma$, so that $\tilde{F}(X_t)=F(\hat{\Sigma}^r_t)$, and use the first equality of (4) at $x=X_u(\omega)$, where $y=\hat{\Sigma}^r_u(\omega)$: the integrand of (3) becomes the integrand displayed in claim 2, as a function of $(u,\omega)$. The measurability assertion for the inner integral is that of conclusion (a) of Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property, the two integrands being the same function.

The rates $q^r$. For $y\neq y'$ in $\mathbb{G}_N$ the map $u\mapsto q^r_u(y,y')$ is either the constant $0$ or $u\mapsto Ny^{\sigma}\beta(\sigma,\gamma,y,a^r(u))$, a scalar multiple of a map shown measurable in Step 3; so it is measurable. Since there are $l(l-1)$ transition labels, $0\le y^{\sigma}\le1$ for $y\in\mathbb{G}_N\subseteq\Delta^l$ and $\beta\le B$, $\sum_{y'\neq y}q^r_u(y,y')\le\sum_cNy^{\sigma}B\le l(l-1)NB$.

$\nu^{D}$ solves the forward equation. Fix $s\in[0,T)$ and $D\in\mathcal{F}^{\mathrm{sys},r}_s$. For $u\in[s,T]$ and $y\in\mathbb{G}_N$, the events $\{X_u=x\}$, $x\in E$, partition $\Omega$ and $\{\hat{\Sigma}^r_u=y\}$ is the union of those with $\Sigma(x)=y$, so by additivity of $P$

moreover $\{\Sigma^r_u=y\}$ is an event (Step 4) that differs from $\{\hat{\Sigma}^r_u=y\}$ only inside $\Omega\setminus\Omega^r$ (Step 1), a set of probability zero, so $P(D\cap\{\hat{\Sigma}^r_u=y\})=P(D\cap\{\Sigma^r_u=y\})$ by monotonicity and additivity of $P$. Property (i) of a solution: $u\mapsto\nu^{D}_u(y)$ is a finite sum of the maps $u\mapsto\mu^{D}_u(x)$, each bounded and $\mathcal{B}_{[s,T]}$-measurable by property (i) for $\mu^{D}$, hence bounded and measurable. Property (ii): let $F:\mathbb{G}_N\to\mathbb{R}$ and write $\mathcal{L}^r_u$ for the operator of Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set for the rates $q^r$ on $\mathbb{G}_N$ and $\mathcal{L}^{E}_u$ for the one for the rates $q^{E}$ on $E$. Grouping the points of $E$ according to $\Sigma(x)$ gives $\nu^{D}_t(F)=\sum_{x\in E}\mu^{D}_t(x)F(\Sigma(x))=\mu^{D}_t(F\circ\Sigma)$ for every $t\in[s,T]$, and likewise $\nu^{D}_u(\mathcal{L}^r_uF)=\mu^{D}_u((\mathcal{L}^r_uF)\circ\Sigma)$. By (4), for every $x\in E$,

the middle step regrouping the labels $e$ by the value $\phi^{e}(x)$ and discarding those with $\phi^{e}(x)=x$, whose terms vanish. Hence $\nu^{D}_u(\mathcal{L}^r_uF)=\mu^{D}_u(\mathcal{L}^{E}_u(F\circ\Sigma))$, and property (ii) for $\mu^{D}$ with the function $F\circ\Sigma$ yields, for every $t\in[s,T]$,

Thus $(\nu^{D}_u)_{u\in[s,T]}$ is a solution of the forward equation on $[s,T]$ for the rates $q^r$ with rate bound $l(l-1)NB$, which completes the proof of claim 2.