Proof of Law Identity between the Reconstructed Record-Frozen Aggregate and the Aggregate Recursion Driven by the Same Control Path

Throughout, $E=\{1,\dots,l\}^N$, $\Sigma:E\to\mathbb{G}_N$ and $X=(X_t)_{t\in[0,T]}$ are as in Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics, so that $\hat{\Sigma}^r_t=\Sigma(X_t)$, and $\mathcal{B}_{[0,T]}$ denotes the trace Borel $\sigma$-algebra on $[0,T]$, with the analogous notation for other compact intervals. We use freely that two events differing only inside an event of probability zero have the same probability (claims 1 and 2 of Basic Properties of a Measure), and that the preimages of sets under a map commute with complements and countable unions, so that the family of sets whose preimage lies in a given $\sigma$-algebra is a $\sigma$-algebra; consequently a map into a generated $\sigma$-algebra is measurable as soon as the preimages of the generators are measurable.

Step 0 (The copy setting is well posed). By claim 1 of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics, $a^r$ is a control path with horizon $T$: its values lie in $\mathcal{A}$ and its components are measurable with respect to $\mathcal{B}_{[0,T]}$. Hence $a^\sharp$ is a control path with horizon $T'$: the preimage under a component of $a^\sharp$ of a Borel set $S'$ of the real line is the intersection with $[0,T']$ of the preimage under the corresponding component of $a^r$, a set of the form $S\cap[0,T]$ with $S$ Borel, and $S\cap[0,T]\cap[0,T']=S\cap[0,T']\in\mathcal{B}_{[0,T']}$. Thus the setting of Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon with horizon $T'$, control path $a^\sharp$, initial point $x_0$, clock horizon $R>NBT'$ and $P^\sharp(\Omega^\sharp_0)=1$ is as required there, and its conclusions are available on $[0,T']$; we write $q^\sharp_u(y,y')$ ($u\in[0,T']$, $y\neq y'$ in $\mathbb{G}_N$) for the rates $q$ of that lemma and $\Sigma^{\mathrm{rec},\sharp}_t(\omega)$ for the recursion path. Moreover $\bar{\Sigma}^\sharp_0=x_0$ on all of $\Omega^\sharp$: in the aggregate recursion of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, $\theta_0=0<T'$ and $x^{(0)}=x_0\in\mathbb{G}_N$, so the recursion does not stop at index $0$, $\theta_1>\theta_0$ by claim 1 of that theorem, and the recursion path satisfies $\Sigma^{\mathrm{rec},\sharp}_0=x^{(0)}=x_0$; as $x_0\in\mathbb{G}_N$, the regularised path satisfies $\bar{\Sigma}^\sharp_0=x_0$.

Step 1 (The rates agree and solutions restrict). By Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon, $q^\sharp_u(y,y')=Ny^{\sigma}\beta(\sigma,\gamma,y,a^\sharp_u)$ if $y'=y+\frac1Nv_c$ with $c=(\sigma,\gamma)$ and $q^\sharp_u(y,y')=0$ otherwise, while by Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics the rates $q^r_u(y,y')$ are given by the same formula with $a^r(u)$ in place of $a^\sharp_u$; since $a^\sharp_u=a^r(u)$ for $u\in[0,T']$, we have $q^\sharp_u=q^r_u$ for every $u\in[0,T']$, and both families have the rate bound $l(l-1)NB$ by those lemmas. Consequently, for $F:\mathbb{G}_N\to\mathbb{R}$, the operators $\mathcal{L}_uF$ of Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set formed with $q^r$ and with $q^\sharp$ coincide for $u\in[0,T']$. We claim: if $(\nu_u)_{u\in[s,T]}$ is a solution of the forward equation on $[s,T]$ for the rates $q^r$ (horizon $T$), with $s<T'$, then $(\nu_u)_{u\in[s,T']}$ is a solution of the forward equation on $[s,T']$ for the rates $q^\sharp$ (horizon $T'$). Indeed, property (i): the restriction to $[s,T']$ of a bounded $\mathcal{B}_{[s,T]}$-measurable map is bounded and $\mathcal{B}_{[s,T']}$-measurable, its preimages being the intersections with $[s,T']$ of the preimages of the original map, which are of the form $S\cap[s,T]$ with $S$ Borel; property (ii): for $t\in[s,T']$ the identity required on $[s,T']$ is the identity of property (ii) on $[s,T]$ at the same $t$, the two integrands $u\mapsto\nu_u(\mathcal{L}_uF)$ agreeing on $[s,t]$ because the operators agree for $u\le T'$. Also, if $(\mu_u)_{u\in[s,T']}$ is a solution on $[s,T']$ for $q^\sharp$ and $\kappa\ge0$ is a real number, then $(\kappa\mu_u)_{u\in[s,T']}$ is a solution: (i) is preserved under multiplication by a constant (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); for (ii), $(\kappa\mu_t)(F)=\kappa\,\mu_t(F)$ and $(\kappa\mu_u)(\mathcal{L}_uF)=\kappa\,\mu_u(\mathcal{L}_uF)$ by the definition of the pairing, and $\int_{[s,t]}\kappa\,\mu_u(\mathcal{L}_uF)\,du=\kappa\int_{[s,t]}\mu_u(\mathcal{L}_uF)\,du$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the integrand being integrable on $[s,t]$ as noted in Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set; so multiplying the identity of (ii) by $\kappa$ gives (ii) for $(\kappa\mu_u)$.

Step 2 (Events of the two filtrations). For $u\in[0,T]$ and $y\in\mathbb{G}_N$, $\{\hat{\Sigma}^r_u=y\}=\bigcup_{x\in E:\,\Sigma(x)=y}\{X_u=x\}$, a finite union of members of $\mathcal{F}^{\mathrm{sys},r}_u$ by (D3) of claim 1 of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics; so $\{\hat{\Sigma}^r_u=y\}\in\mathcal{F}^{\mathrm{sys},r}_u\subseteq\mathcal{F}$, and in particular $D_0\in\mathcal{F}^{\mathrm{sys},r}_0$. Likewise, for $u\in[0,T']$, $\{\bar{\Sigma}^\sharp_u=y\}\in\mathfrak{F}^\sharp_u\subseteq\mathcal{F}^\sharp$ by (D3) of claim 1 of Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon (the state space there being $\mathbb{G}_N$, not the $E$ of this proof). Since the two families are filtrations, events of $\mathcal{F}^{\mathrm{sys},r}_{u}$ belong to $\mathcal{F}^{\mathrm{sys},r}_{u'}$ for $u\le u'$, and likewise for $\mathfrak{F}^\sharp$.

Step 3 (Claim 1, by induction). Fix $n$, times $0\le s_1<\dots<s_n\le T'$ and points $x_1,\dots,x_n\in\mathbb{G}_N$; put $s_0=0$ and, for $j\in\{0,\dots,n\}$,

$$D_j=D_0\cap\bigcap_{i=1}^{j}\{\hat{\Sigma}^r_{s_i}=x_i\},\qquad D^\sharp_j=\bigcap_{i=1}^{j}\{\bar{\Sigma}^\sharp_{s_i}=x_i\}\quad(D^\sharp_0=\Omega^\sharp).$$

By Step 2, $D_j\in\mathcal{F}^{\mathrm{sys},r}_{s_j}$ and $D^\sharp_j\in\mathfrak{F}^\sharp_{s_j}$ (for $j=0$ because $\Omega^\sharp\in\mathfrak{F}^\sharp_0$, a $\sigma$-algebra on $\Omega^\sharp$). We prove by induction on $j$ the statement

$$(H_j)\qquad P\bigl(D_j\cap\{\hat{\Sigma}^r_u=y\}\bigr)=P(D_0)\,P^\sharp\bigl(D^\sharp_j\cap\{\bar{\Sigma}^\sharp_u=y\}\bigr)\quad\text{for all }u\in[s_j,T']\text{ and }y\in\mathbb{G}_N .$$

Let $j\in\{0,\dots,n\}$ and assume $(H_{j-1})$ if $j\ge1$. First suppose $s_j=T'$ (so $j\ge1$, since $s_0=0<T'$); then $u=T'=s_j$, and $D_j\cap\{\hat{\Sigma}^r_{s_j}=y\}$ equals $D_{j-1}\cap\{\hat{\Sigma}^r_{s_j}=x_j\}$ if $y=x_j$ and is empty otherwise, and likewise $D^\sharp_j\cap\{\bar{\Sigma}^\sharp_{s_j}=y\}$ equals $D^\sharp_{j-1}\cap\{\bar{\Sigma}^\sharp_{s_j}=x_j\}$ if $y=x_j$ and is empty otherwise; so $(H_j)$ follows from $(H_{j-1})$ at $u=s_j\in[s_{j-1},T']$ and $y=x_j$. Now suppose $s_j<T'$. By claim 2 of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics with $s=s_j\in[0,T)$ and $D=D_j$, the family $\nu_u(y)=P(D_j\cap\{\hat{\Sigma}^r_u=y\})$, $u\in[s_j,T]$, is a solution of the forward equation on $[s_j,T]$ for the rates $q^r$, hence, by Step 1, $(\nu_u)_{u\in[s_j,T']}$ is a solution on $[s_j,T']$ for the rates $q^\sharp$. By claim 2 of Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon (horizon $T'$) with its time $r$ equal to $s_j\in[0,T')$ and $D=D^\sharp_j$, the family $\mu_u(y)=P^\sharp(D^\sharp_j\cap\{\bar{\Sigma}^\sharp_u=y\})$, $u\in[s_j,T']$, is a solution on $[s_j,T']$ for $q^\sharp$, and so is $(P(D_0)\mu_u)_{u\in[s_j,T']}$ by Step 1. The two solutions agree at $u=s_j$. If $j=0$: $\nu_0(y)=P(D_0\cap\{\hat{\Sigma}^r_0=y\})$ equals $P(D_0)$ if $y=x_0$ and $0$ otherwise, while $\mu_0(y)=P^\sharp(\bar{\Sigma}^\sharp_0=y)$ equals $1$ if $y=x_0$ and $0$ otherwise by Step 0. If $j\ge1$: $\nu_{s_j}(y)=P(D_j\cap\{\hat{\Sigma}^r_{s_j}=y\})$ equals $P(D_{j-1}\cap\{\hat{\Sigma}^r_{s_j}=x_j\})$ if $y=x_j$ and $0$ otherwise, which by $(H_{j-1})$ at $u=s_j\in[s_{j-1},T']$ equals $P(D_0)P^\sharp(D^\sharp_{j-1}\cap\{\bar{\Sigma}^\sharp_{s_j}=x_j\})=P(D_0)P^\sharp(D^\sharp_j)$ if $y=x_j$ and $0$ otherwise; and $P(D_0)\mu_{s_j}(y)=P(D_0)P^\sharp(D^\sharp_j\cap\{\bar{\Sigma}^\sharp_{s_j}=y\})$ equals $P(D_0)P^\sharp(D^\sharp_j)$ if $y=x_j$ and $0$ otherwise. Hence, by Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set applied on the finite set $\mathbb{G}_N$ with horizon $T'$, time $s_j\in[0,T')$, rates $q^\sharp$ and rate bound $l(l-1)NB$, $\nu_u=P(D_0)\mu_u$ for every $u\in[s_j,T']$, which is $(H_j)$. This completes the induction. Claim 1 is $(H_n)$ at $u=s_n$ and $y=x_n$: there $D_n\cap\{\hat{\Sigma}^r_{s_n}=x_n\}=D_n$ and $D^\sharp_n\cap\{\bar{\Sigma}^\sharp_{s_n}=x_n\}=D^\sharp_n$.

Step 4 (The maps $\Pi^r$ and $\Pi^\sharp$). Let $\omega\in\Omega^r$. By (H1) of claim 1 of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics, there are a count $K$ and times $0<t_1<\dots<t_K\le T$ such that $t\mapsto X_t(\omega)$ is constant on $[0,t_1)$, on each $[t_i,t_{i+1})$ and on $[t_K,T]$ (on $[0,T]$ if $K=0$); the same then holds for $t\mapsto\hat{\Sigma}^r_t(\omega)=\Sigma(X_t(\omega))$. Let $k$ be the number of indices $i$ with $t_i\le T'$. Then the restriction of $\hat{\Sigma}^r_\cdot(\omega)$ to $[0,T']$ is constant on $[0,t_1)$ if $k\ge1$, on $[t_i,t_{i+1})$ for $i<k$, and on $[t_k,T']$, since $[t_k,T']\subseteq[t_k,t_{k+1})$ when $k<K$ and $[t_k,T']\subseteq[t_K,T]$ when $k=K$; if $k=0$ it is constant on $[0,T']\subseteq[0,t_1)$ (or on $[0,T']\subseteq[0,T]$ when $K=0$). Hence $\Pi^r(\omega)\in\mathsf{Path}$ with the times $t_1,\dots,t_k$; for $\omega\notin\Omega^r$, $\Pi^r(\omega)$ is constant, so lies in $\mathsf{Path}$ with $k=0$. For $\omega\in\Omega^\sharp_0$ the data $(\mathsf{P}^\sharp(\omega),a^\sharp,x_0)$ are conflict-free, so by claim 2 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution the recursion path $t\mapsto\Sigma^{\mathrm{rec},\sharp}_t(\omega)$ is an open-loop aggregate solution on $[0,T']$; condition 1 of that definition says precisely that it takes values in $\mathbb{G}_N$ and belongs to $\mathsf{Path}$, and $\bar{\Sigma}^\sharp_\cdot(\omega)=\Sigma^{\mathrm{rec},\sharp}_\cdot(\omega)$ because the path stays in $\mathbb{G}_N$. For $\omega\notin\Omega^\sharp_0$, $\Pi^\sharp(\omega)$ is constant. Thus both maps take values in $\mathsf{Path}$.

For measurability it suffices, by the remark in the preamble, to consider a generator $Z=\{p\in\mathsf{Path}:p(u)=y\}$ with $u\in[0,T']$ and $y\in\mathbb{G}_N$. We have $\{\Pi^r\in Z\}=(\Omega^r\cap\{\hat{\Sigma}^r_u=y\})\cup((\Omega\setminus\Omega^r)\cap\{\omega\in\Omega:x_0=y\})$, the last set being $\Omega\setminus\Omega^r$ if $y=x_0$ and empty otherwise; both pieces lie in $\mathcal{F}$ by Step 2 and $\Omega^r\in\mathcal{F}$. Likewise $\{\Pi^\sharp\in Z\}=(\Omega^\sharp_0\cap\{\bar{\Sigma}^\sharp_u=y\})\cup((\Omega^\sharp\setminus\Omega^\sharp_0)\cap\{\omega\in\Omega^\sharp:x_0=y\})\in\mathcal{F}^\sharp$, since $\Omega^\sharp_0\in\mathcal{F}^\sharp$ (claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, as recalled in Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon). Hence $\Pi^r$ and $\Pi^\sharp$ are measurable.

Step 5 (Cylinder sets). For $k\ge1$, times $0\le u_1<\dots<u_k\le T'$ and points $y_1,\dots,y_k\in\mathbb{G}_N$ let $Z(u_1,\dots,u_k;y_1,\dots,y_k)=\{p\in\mathsf{Path}:p(u_i)=y_i\text{ for }i=1,\dots,k\}$ (a cylinder set), and let $\mathcal{Z}$ be the family of all cylinder sets together with the empty set. $\mathcal{Z}$ is a $\pi$-system: it is nonempty, and the intersection of two cylinder sets is either empty (when they prescribe different values at a common time) or the cylinder set on the increasing enumeration of the union of their two time sets with the prescribed values (which agree at common times). The generators of $\mathcal{C}$ are the cylinder sets with $k=1$, so $\mathcal{C}$ is contained in the $\sigma$-algebra generated by $\mathcal{Z}$; conversely every cylinder set is a finite intersection of generators, so $\mathcal{Z}\subseteq\mathcal{C}$ and the $\sigma$-algebra generated by $\mathcal{Z}$ is contained in $\mathcal{C}$ (Generated Sigma-Algebra). Hence $\mathcal{C}$ is the $\sigma$-algebra generated by $\mathcal{Z}$.

For a cylinder set $Z=Z(u_1,\dots,u_k;y_1,\dots,y_k)$ we have $\{\Pi^r\in Z\}\cap\Omega^r=\bigcap_{i=1}^k\{\hat{\Sigma}^r_{u_i}=y_i\}\cap\Omega^r$ and $\{\Pi^\sharp\in Z\}\cap\Omega^\sharp_0=\bigcap_{i=1}^k\{\bar{\Sigma}^\sharp_{u_i}=y_i\}\cap\Omega^\sharp_0$, so that, $\Omega\setminus\Omega^r$ and $\Omega^\sharp\setminus\Omega^\sharp_0$ having probability zero,

$$P\bigl(D_0\cap\{\Pi^r\in Z\}\bigr)=P\Bigl(D_0\cap\bigcap_{i=1}^k\{\hat{\Sigma}^r_{u_i}=y_i\}\Bigr)=P(D_0)\,P^\sharp\Bigl(\bigcap_{i=1}^k\{\bar{\Sigma}^\sharp_{u_i}=y_i\}\Bigr)=P(D_0)\,P^\sharp\bigl(\Pi^\sharp\in Z\bigr),\qquad(1)$$

the middle equality being claim 1 with $n=k$; (1) holds trivially for $Z=\emptyset$.

Step 6 (Claim 2). Let $P_0$ be the measure with density $\mathbf{1}_{D_0}$ with respect to $P$ on $(\Omega,\mathcal{F})$ (claim 3 of that lemma): for $A\in\mathcal{F}$, $P_0(A)=\int_\Omega\mathbf{1}_A\mathbf{1}_{D_0}\,dP=P(A\cap D_0)$, the integrand being the nonnegative simple function $\mathbf{1}_{A\cap D_0}$. Let $Q=(P_0)_{\Pi^r}$ and $Q^\sharp=(P^\sharp)_{\Pi^\sharp}$ be the image measures on $(\mathsf{Path},\mathcal{C})$ (claim 1 of that lemma, $\Pi^r$ and $\Pi^\sharp$ being measurable by Step 4), so that $Q(A)=P(D_0\cap\{\Pi^r\in A\})$ and $Q^\sharp(A)=P^\sharp(\Pi^\sharp\in A)$ for $A\in\mathcal{C}$, with $Q(\mathsf{Path})=P(D_0)$ and $Q^\sharp(\mathsf{Path})=1$. Let $\tilde{Q}$ be the measure with density $\varrho\equiv P(D_0)$ with respect to $Q^\sharp$: $\tilde{Q}(A)=\int_{\mathsf{Path}}\mathbf{1}_A\,\varrho\,dQ^\sharp=P(D_0)\,Q^\sharp(A)$ for $A\in\mathcal{C}$, the integrand being the nonnegative simple function $P(D_0)\mathbf{1}_A$. Then $Q$ and $\tilde{Q}$ are measures on $\mathcal{C}$ with $Q(\mathsf{Path})=P(D_0)=\tilde{Q}(\mathsf{Path})<\infty$, and by (1) they agree on the $\pi$-system $\mathcal{Z}$, which generates $\mathcal{C}$ by Step 5. By claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, $Q=\tilde{Q}$, that is, $P(D_0\cap\{\Pi^r\in A\})=P(D_0)P^\sharp(\Pi^\sharp\in A)$ for every $A\in\mathcal{C}$, the first identity of claim 2.

Finally let $\Phi:\mathsf{Path}\to[0,\infty]$ be $\mathcal{C}$-measurable. The compositions $\Phi\circ\Pi^r$ and $\Phi\circ\Pi^\sharp$ are measurable in the sense of Lebesgue Integral of a Nonnegative Measurable Function, since $\{\Phi\circ\Pi^r>\alpha\}=\{\Pi^r\in\{\Phi>\alpha\}\}$ and $\{\Phi>\alpha\}\in\mathcal{C}$ for every real $\alpha$, and likewise for $\Pi^\sharp$. Using successively claim 3 of Image Measures, Measures with Densities, and Change of Variables for $P_0$, claim 2 of that lemma for $Q$, the identity $Q=\tilde{Q}$, claim 3 of that lemma for $\tilde{Q}$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral with the constant $P(D_0)\in[0,\infty)$, and claim 2 of Image Measures, Measures with Densities, and Change of Variables for $Q^\sharp$,

$$\int_\Omega\mathbf{1}_{D_0}\,\Phi(\Pi^r)\,dP=\int_\Omega\Phi\circ\Pi^r\,dP_0=\int_{\mathsf{Path}}\Phi\,dQ=\int_{\mathsf{Path}}\Phi\,d\tilde{Q}=\int_{\mathsf{Path}}\Phi\,\varrho\,dQ^\sharp=P(D_0)\int_{\mathsf{Path}}\Phi\,dQ^\sharp=P(D_0)\int_{\Omega^\sharp}\Phi\circ\Pi^\sharp\,dP^\sharp ,$$

which is the second identity of claim 2.