Conditional Restart of the Record Channel at an Intermediate Time

Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled $N$-agent dynamics with transition-rate family $\beta$ with rate bound $B$, observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$, horizon $T>0$, $N$-agent driving system $(\Omega,\mathcal{F},P)$, observation-driven control policy $h=(h_k)_{k\ge0}$ with control dimension $m$, and a solution on $[0,T]$ with regular event $\Omega_0$, state processes $\sigma^i$, observation processes $\Upsilon^\upsilon$, control $\alpha$, empirical state measure $\Sigma_t$, consumed clock times $A^{i,\sigma\gamma}_t$ and $\tilde{A}^{i,\upsilon}_t$, counters $N^{i,\sigma\gamma}_t$ and $\tilde{N}^{i,\upsilon}_t$, observation-event count $K_t$, event times $\tau_j$ with channels $\upsilon_j$, and filtrations $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$; the observation record space $(\mathbf{R}_T,\mathcal{R}_T,\rho_T)$ (horizon-$a$ spaces written $(\mathbf{R}_a,\mathcal{R}_a,\rho_a)$); the $\sigma$-algebra $\mathcal{T}$; fixed reconstruction data $(\eta^{r,i,\gamma},\sigma^{r,i},\tilde{A}^{r,i,\upsilon},G)$ with reconstructed empirical state measures $\Sigma^r$; the record density kernel $f$; the observation record $W$; and the aggregate observation drift $\tilde{b}$ with total observation rate $\tilde{b}^{\mathrm{tot}}$.

Fix $s\in(0,T)$. Let $\hat{Y}^{i,\sigma\gamma}$ and $\hat{\tilde{Y}}^{i,\upsilon}$ be the residual clocks at time $s$, let $\pi_s$, $\iota_s$, $\oplus_s$ be the prefix, increment, and concatenation maps at $s$, and put $W_s=\pi_s(W)$ and $\hat{W}=\iota_s(W)$. Define the segment processes on $[0,T-s]$ by $\hat{\sigma}^i_u=\sigma^i_{s+u}$, $\hat{\Upsilon}^\upsilon_u=\Upsilon^\upsilon_{s+u}-\Upsilon^\upsilon_s$, and $\hat{\alpha}_u=\alpha_{s+u}$. Let $\mathcal{T}_s$ be the $\sigma$-algebra generated by $\mathcal{F}^{\mathrm{sys}}_s$ together with the residual transition-clock variables $\hat{Y}^{i,\sigma\gamma}_u$ for all indices and all $u\ge0$.

1. (Restarted system and prefix-frozen policies) The space $(\Omega,\mathcal{F},P)$ with initial states $\sigma^1_s,\dots,\sigma^N_s$ and the residual clocks is an $N$-agent driving system by Fresh-Start Property of the Controlled N-Agent Dynamics, called the restarted system. For $\varrho=(\kappa,\mathbf{t},w)\in\mathbf{R}_s$, with time entries $\mathbf{t}_1,\dots,\mathbf{t}_\kappa$ (a symbol chosen to avoid the solution's event times $\tau_j$), the prefix-frozen policy $h^{s,\varrho}=((h^{s,\varrho})_j)_{j\ge0}$, defined by $(h^{s,\varrho})_j(u,\tau',w')=h_{\kappa+j}\bigl(s+u,\ (\mathbf{t}_1,\dots,\mathbf{t}_\kappa,\ s+\tau'_1,\dots,s+\tau'_j),\ (w_1,\dots,w_\kappa,\ w'_1,\dots,w'_j)\bigr)$ for $j\ge1$, $u\in[0,T-s]$, $\tau'$ in the set of $j$-tuples of Observation-Driven Control Policy with horizon $T-s$, and channel tuples $w'$, and by $(h^{s,\varrho})_0(u)=h_\kappa(s+u,\mathbf{t},w)$, is an observation-driven control policy with horizon $T-s$, control dimension $m$, and $\tilde{l}$ channels. Moreover $W_s^{-1}(A)\in\mathcal{G}_s$ for every $A\in\mathcal{R}_s$, and on $\Omega_0$ the value $W_s$ equals the observation record of the horizon-$s$ restricted solution of claim 4 of Bayes Disintegration and Filtering Formula for the Observation Record.

2. (Clock recovery and independence) $\mathcal{T}\subseteq\mathcal{T}_s$; and the $\sigma$-algebra generated by the residual observation-clock variables together with the events of probability zero is independent of $\mathcal{T}_s$.

3. (Segment conditions and the prefix-frozen control identity) There is an event $\Omega^*_s\subseteq\Omega_0$ with $P(\Omega^*_s)=1$ such that: (i) at every $\omega\in\Omega^*_s$, every map $t\mapsto N^{i,\sigma\gamma}_t(\omega)$ and every map $t\mapsto\tilde{N}^{i,\upsilon}_t(\omega)$ is continuous at $t=s$; (ii) the segment processes together with the regular event $\Omega^*_s$ satisfy conditions 1, 2, 3, 4, and 6 of Solution of the Controlled N-Agent Dynamics on $[0,T-s]$ for the restarted system, whose consumed clock times and counters agree on $\Omega^*_s$ with $\hat{A}^{i,\sigma\gamma}_u=A^{i,\sigma\gamma}_{s+u}-A^{i,\sigma\gamma}_s$, $\hat{\tilde{A}}^{i,\upsilon}_u=\tilde{A}^{i,\upsilon}_{s+u}-\tilde{A}^{i,\upsilon}_s$, $\hat{N}^{i,\sigma\gamma}_u=N^{i,\sigma\gamma}_{s+u}-N^{i,\sigma\gamma}_s$, and $\hat{\tilde{N}}^{i,\upsilon}_u=\tilde{N}^{i,\upsilon}_{s+u}-\tilde{N}^{i,\upsilon}_s$, and vanish off $\Omega^*_s$; (iii) at every $\omega\in\Omega^*_s$, writing $\hat{K}_u=K_{s+u}-K_s$, $\hat{\tau}_j=\tau_{K_s+j}-s$, and $\hat{\upsilon}_j=\upsilon_{K_s+j}$, the control identity of condition 5 of Solution of the Controlled N-Agent Dynamics holds in the form $\hat{\alpha}_u(\omega)=(h^{s,W_s(\omega)})_{\hat{K}_u(\omega)}\bigl(u,\ \hat{\tau}_1(\omega),\dots,\hat{\tau}_{\hat{K}_u}(\omega),\ \hat{\upsilon}_1(\omega),\dots,\hat{\upsilon}_{\hat{K}_u}(\omega)\bigr)\qquad(u\in[0,T-s]);$ in particular, for every $\varrho\in\mathbf{R}_s$, at every $\omega\in\Omega^*_s$ with $W_s(\omega)=\varrho$, all conditions 1--6 hold at $\omega$ with the fixed policy $h^{s,\varrho}$; (iv) on $\Omega^*_s$, the observation events of the segment are the times $\hat{\tau}_j$ with channels $\hat{\upsilon}_j$, and the observation record formed from them equals $\hat{W}=\iota_s(W)$.

4. (Restart kernel and factorization) Define $\hat{f}:\mathbf{R}_{T-s}\times\Omega\to[0,\infty)$ by $\hat{f}(r',\omega)=0$ when $(W_s(\omega)\oplus_s r',\omega)\notin G$ and otherwise, writing $r'=(k',t',v')$ and $q=W_s(\omega)\oplus_s r'$, $\hat{f}(r',\omega)=\Bigl(\prod_{j=1}^{k'}N\,\tilde{b}^{v'_j}\bigl(\Sigma^{q}_{(s+t'_j)-}(\omega)\bigr)\Bigr)\exp\Bigl(-N\int_{[s,T]}\tilde{b}^{\mathrm{tot}}\bigl(\Sigma^{q}_u(\omega)\bigr)\,du\Bigr),$ with the empty product equal to $1$; and define the prefix factor $f_s:\Omega\to[0,\infty)$ as $0$ off $\Omega_0$ and, on $\Omega_0$, $f_s=\Bigl(\prod_{j=1}^{K_s}N\,\tilde{b}^{\upsilon_j}\bigl(\Sigma_{\tau_j-}\bigr)\Bigr)\exp\Bigl(-N\int_{[0,s]}\tilde{b}^{\mathrm{tot}}\bigl(\Sigma_u\bigr)\,du\Bigr),$ where $\Sigma_{\tau_j-}$ is the left limit at $\tau_j$ of the empirical state path, which exists since the state paths are piecewise constant ($f_s$ enters the claims below only through the pointwise factorization display). Then $\hat{f}$ is measurable with respect to $\mathcal{R}_{T-s}\otimes\mathcal{T}_s$ (product $\sigma$-algebra); on the cell of $\mathbf{R}_{T-s}$ with $k'$ events, $0\le\hat{f}\le(N\tilde{B})^{k'}$; and there is an event of probability one on which, simultaneously for every $r'\in\mathbf{R}_{T-s}$, $f\bigl(W_s\oplus_s r',\cdot\bigr)=f_s\cdot\hat{f}(r',\cdot).$

5. (Conditional density of the increment record) For every $\mathcal{T}_s$-measurable $Z:\Omega\to[0,\infty]$ and every $\mathcal{R}_{T-s}$-measurable $g:\mathbf{R}_{T-s}\to[0,\infty]$, $\mathbb{E}\bigl[Z\,g(\hat{W})\bigr]=\mathbb{E}\Bigl[Z\int_{\mathbf{R}_{T-s}}g(r')\,\hat{f}(r',\cdot)\,\rho_{T-s}(dr')\Bigr]\qquad\text{in }[0,\infty],$ with expectations of $[0,\infty]$-valued maps understood as integrals with respect to $P$; in particular, almost surely $\int_{\mathbf{R}_{T-s}}\hat{f}(r',\cdot)\,\rho_{T-s}(dr')=1$.