Frontier-Window and Crossing-Compensation Identities for Jointly Driven Solutions of the Controlled N-Agent Dynamics

Adopt the setting, notation, and hypotheses of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics: the probability space $(\Omega,\mathcal{F},P)$ with transition and observation clocks and the $\sigma$-algebra $\mathcal{S}_0$; the clock labels $a$ with clocks $Y^{a}$ and rate bounds $B_a$; and, for $j\in\{1,\dots,J\}$, the $\mathcal{S}_0$-measurable initial states $\varsigma_j$, the policies $h_j$, and the given solutions of the controlled $N$-agent dynamics on $[0,T]$, with consumed clock times $A_j^{a}$ and system filtrations $(\mathcal{F}^{\mathrm{sys},j}_t)_{t\in[0,T]}$. For $t\in[0,T]$ let $\mathbb{F}_t=\mathcal{F}^{\mathrm{sys},1}_t\vee\dots\vee\mathcal{F}^{\mathrm{sys},J}_t$ be the $\sigma$-algebra generated by the union, and for a clock label $a$ define the frontier consumed time $A^{\vee,a}_t=\max(A_1^{a}(t),\dots,A_J^{a}(t))$. Let $\mu_p$ and $\exp$ be as in Predictable-Window Moment Identities for the Homogeneous Poisson Process, and write $\mathbb{E}$ for the expectation, expectations of $[0,\infty]$-valued random variables being their integrals with respect to $P$.

1. (Frontier windows) Let $t\in[0,T]$, let $a$ be a clock label, let $\bar{V}$ and $\bar{\Lambda}$ be nonnegative reals, and let $v$ and $\lambda$ be $\mathbb{F}_t$-measurable random variables with $A^{\vee,a}_t(\omega)\le v(\omega)\le\bar{V}$ and $0\le\lambda(\omega)\le\bar{\Lambda}$ for every $\omega$. Then for every natural number $p\ge1$ and every $\mathbb{F}_t$-measurable $Z:\Omega\to[0,\infty]$, in $[0,\infty]$: $\mathbb{E}\bigl[Z\,\bigl(Y^{a}_{v+\lambda}-Y^{a}_{v}\bigr)^{p}\bigr]=\mathbb{E}\bigl[Z\,\mu_p(\lambda)\bigr]\qquad\text{and}\qquad \mathbb{E}\bigl[Z\,\mathbf{1}\{Y^{a}_{v+\lambda}-Y^{a}_{v}\ge1\}\bigr]=\mathbb{E}\bigl[Z\,\bigl(1-\exp(-\lambda)\bigr)\bigr]\le\mathbb{E}\bigl[Z\,\lambda\bigr];$ in particular, taking $p=1$, $\mathbb{E}[Z\,(Y^{a}_{v+\lambda}-Y^{a}_{v})]=\mathbb{E}[Z\,\lambda]$.

2. (Crossing compensation) For all real $0\le s\le w\le T$, every clock label $a$, and every $\mathbb{F}_s$-measurable $Z:\Omega\to[0,\infty]$, in $[0,\infty]$: $\mathbb{E}\bigl[Z\,\bigl(Y^{a}_{A^{\vee,a}_w}-Y^{a}_{A^{\vee,a}_s}\bigr)\bigr]=\mathbb{E}\bigl[Z\,\bigl(A^{\vee,a}_w-A^{\vee,a}_s\bigr)\bigr].$

3. (Product windows) Let $t\in[0,T]$, let $a_1,\dots,a_k$ be distinct clock labels with $k\ge1$ a natural number, let $\bar{V}$ and $\bar{\Lambda}$ be nonnegative reals, and for each $n\in\{1,\dots,k\}$ let $v_n$ and $\lambda_n$ be $\mathbb{F}_t$-measurable random variables with $A^{\vee,a_n}_t(\omega)\le v_n(\omega)\le\bar{V}$ and $0\le\lambda_n(\omega)\le\bar{\Lambda}$ for every $\omega$. Then for all natural numbers $p_1,\dots,p_k\ge1$ and every $\mathbb{F}_t$-measurable $Z:\Omega\to[0,\infty]$, in $[0,\infty]$ and with the finite product notation: $\mathbb{E}\Bigl[Z\,\prod_{n=1}^{k}\bigl(Y^{a_n}_{v_n+\lambda_n}-Y^{a_n}_{v_n}\bigr)^{p_n}\Bigr]=\mathbb{E}\Bigl[Z\,\prod_{n=1}^{k}\mu_{p_n}(\lambda_n)\Bigr];$ moreover the identity continues to hold when, for the indices $n$ in an arbitrary subset of $\{1,\dots,k\}$, the factor $\bigl(Y^{a_n}_{v_n+\lambda_n}-Y^{a_n}_{v_n}\bigr)^{p_n}$ is replaced by $\mathbf{1}\{Y^{a_n}_{v_n+\lambda_n}-Y^{a_n}_{v_n}\ge1\}$ and correspondingly $\mu_{p_n}(\lambda_n)$ by $1-\exp(-\lambda_n)$.