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

Adopt the setting and notation of Forward Equation on the Aggregate Lattice for the Reconstructed Record-Frozen N-Agent Dynamics with its fixed record $r\in\mathbf{R}$: in particular the natural numbers $N\ge1$, $l\ge2$ and $m\ge1$, the control set $\mathcal{A}\subseteq\mathbb{R}^m$, the transition-rate family $\beta$ with rate bound $B$, the horizon $T>0$, the probability space $(\Omega,\mathcal{F},P)$ of the $N$-agent driving system, the record-frozen control path $a^r$, the event $\Omega^r$ of probability one, the family $\hat{\Sigma}^r=(\hat{\Sigma}^r_t)_{t\in[0,T]}$ of maps with values in the aggregate lattice $\mathbb{G}_N$, and the system filtration $(\mathcal{F}^{\mathrm{sys},r}_t)_{t\in[0,T]}$. Write $\mathbf{1}_S$ for the indicator of a set $S$ and $\{\Pi\in A\}$ for the preimage of a set $A$ under a map $\Pi$.

Let $T'$ be a real number with $0<T'\le T$, let $x_0\in\mathbb{G}_N$, and let $a^\sharp:[0,T']\to\mathcal{A}$ be the restriction of $a^r$ to $[0,T']$, $a^\sharp_u=a^r(u)$ for $u\in[0,T']$, which is a control path with horizon $T'$. Adopt the setting of Forward Equation for the Aggregate Recursion Driven by Independent Poisson Clocks with a Horizon with the same $N$, $l$, $m$, $\mathcal{A}$, $\beta$ and $B$, with the horizon $T'$ in place of $T$, with the control path $a^\sharp$ and the initial point $x_0$, and with the probability space there called $(\Omega^\sharp,\mathcal{F}^\sharp,P^\sharp)$: it carries a family $\mathsf{P}^\sharp=(\mathsf{P}^{\sharp,c})$ of Poisson clocks with a horizon $R>NBT'$ whose generated $\sigma$-algebras are independent, the aggregate recursion of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution for the data $(\mathsf{P}^\sharp(\omega),a^\sharp,x_0)$ is run for every $\omega\in\Omega^\sharp$, and $\bar{\Sigma}^\sharp=(\bar{\Sigma}^\sharp_t)_{t\in[0,T']}$, $(\mathfrak{F}^\sharp_t)_{t\in[0,T']}$ and $\Omega^\sharp_0$ denote the regularised recursion path, the recursion filtration and the conflict-free event of that lemma; as there, assume $P^\sharp(\Omega^\sharp_0)=1$ ($\Omega^\sharp_0$ is the conflict-free event of the horizon-$T'$ recursion; when $T'=T$ the copy clocks of The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them qualify, by claims 2, 3 and 4 of that lemma, provided their horizon satisfies the strict inequality $R>NBT$). Put $D_0=\{\hat{\Sigma}^r_0=x_0\}$, an event of $\mathcal{F}^{\mathrm{sys},r}_0$. Then:

1. (Finite-dimensional laws.) For every natural number $n$, all times $0\le s_1<\dots<s_n\le T'$ and all points $x_1,\dots,x_n\in\mathbb{G}_N$,

$$P\Bigl(D_0\cap\bigcap_{j=1}^{n}\{\hat{\Sigma}^r_{s_j}=x_j\}\Bigr)=P(D_0)\,P^\sharp\Bigl(\bigcap_{j=1}^{n}\{\bar{\Sigma}^\sharp_{s_j}=x_j\}\Bigr).$$

2. (Path laws.) Let $\mathsf{Path}=\mathsf{Path}(\mathbb{G}_N,T')$ be the space of piecewise constant paths in $\mathbb{G}_N$ with horizon $T'$, that is, the set of maps $p:[0,T']\to\mathbb{G}_N$ for which there are a count $k$, either $0$ or a natural number, and times $0<t_1<\dots<t_k\le T'$ such that $p$ is constant on $[0,t_1)$, on $[t_j,t_{j+1})$ for each $j\in\{1,\dots,k-1\}$, and on $[t_k,T']$ (constant on $[0,T']$ when $k=0$), and let $\mathcal{C}=\mathcal{C}_{T'}$ be the $\sigma$-algebra generated on $\mathsf{Path}$ by the sets $\{p\in\mathsf{Path}:p(u)=y\}$ with $u\in[0,T']$ and $y\in\mathbb{G}_N$, as in that lemma. Define $\Pi^r$ on $\Omega$, with values in the set of maps $[0,T']\to\mathbb{G}_N$, by $\Pi^r(\omega)(u)=\hat{\Sigma}^r_u(\omega)$ for $\omega\in\Omega^r$ and $\Pi^r(\omega)(u)=x_0$ for $\omega\notin\Omega^r$ ($u\in[0,T']$), and $\Pi^\sharp$ on $\Omega^\sharp$, with values in the same set of maps, by $\Pi^\sharp(\omega)(u)=\bar{\Sigma}^\sharp_u(\omega)$ for $\omega\in\Omega^\sharp_0$ and $\Pi^\sharp(\omega)(u)=x_0$ for $\omega\notin\Omega^\sharp_0$. Then $\Pi^r$ and $\Pi^\sharp$ take their values in $\mathsf{Path}$ and are measurable with respect to $\mathcal{F}$, respectively $\mathcal{F}^\sharp$, and $\mathcal{C}$; for every $A\in\mathcal{C}$,

$$P\bigl(D_0\cap\{\Pi^r\in A\}\bigr)=P(D_0)\,P^\sharp\bigl(\Pi^\sharp\in A\bigr);$$

and for every function $\Phi:\mathsf{Path}\to[0,\infty]$ that is measurable with respect to $\mathcal{C}$,

$$\int_{\Omega}\mathbf{1}_{D_0}\,\Phi(\Pi^r)\,dP=P(D_0)\int_{\Omega^\sharp}\Phi(\Pi^\sharp)\,dP^\sharp ,$$

both sides being integrals of nonnegative measurable functions with values in $[0,\infty]$, the product $\mathbf{1}_{D_0}\Phi(\Pi^r)$ being formed with the multiplication convention $0\cdot\infty=0$ of Image Measures, Measures with Densities, and Change of Variables.