Below-Frontier Evaluations, Tilted Poisson Moment Functions, and Multi-Base Window Bounds 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 --- in particular the $\sigma$-algebra $\mathcal{S}_0$ and the independence of the finite family consisting of $\mathcal{S}_0$ together with the $\sigma$-algebra generated by the variables of each single clock --- together with the derived objects of Frontier-Window and Crossing-Compensation Identities for Jointly Driven Solutions of the Controlled N-Agent Dynamics (the fixed level $r$, the caps $c_a$, and the $\sigma$-algebra and events of the third paragraph of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics are not used in the present lemma): the probability space $(\Omega,\mathcal{F},P)$ with transition and observation clocks, the clock labels $a$ with clocks $Y^{a}$ and rate bounds $B_a$, the $J$ given solutions of the controlled $N$-agent dynamics on $[0,T]$ with consumed clock times $A_j^{a}$, for $t\in[0,T]$ the $\sigma$-algebra $\mathbb{F}_t$ generated by the union of the $J$ system $\sigma$-algebras at $t$, and for each clock label $a$ 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 $P$. For a natural number $p\ge0$ and a real $\kappa\ge0$ define $\mu_{p,\kappa}:[0,\infty)\to[0,\infty]$ by $\mu_{p,\kappa}(x)\ :=\ \exp(-x)\,\sum_{k\ge0}\frac{k^{p}\,\exp(\kappa k)\,x^{k}}{k!},$ the sum over the natural numbers $k\ge0$, with the factorial $k!$, well defined by nonnegativity as the least upper bound of its finite partial sums, with the conventions $0^{0}=1$ and $k^{0}=1$; all products of $[0,\infty]$-valued quantities below are formed with the convention that zero times infinity is zero.

1. (Below-frontier evaluations) Let $t\in[0,T]$, let $a$ be a clock label, and let $W$ be an $\mathbb{F}_t$-measurable random variable with $0\le W(\omega)\le A^{\vee,a}_t(\omega)$ for every $\omega$. Then the evaluation $Y^{a}_{W}$ of the clock path at the level $W$ is an $\mathbb{F}_t$-measurable random variable.

2. (Tilted product windows) Let $t\in[0,T]$, let $a_1,\dots,a_k$ be distinct clock labels with $k\ge1$ a natural number, and let $\bar{V}$ and $\bar{\Lambda}$ be nonnegative reals. For each $i\in\{1,\dots,k\}$ let $v_i$ and $\lambda_i$ be $\mathbb{F}_t$-measurable random variables with $A^{\vee,a_i}_t(\omega)\le v_i(\omega)\le\bar{V}$ and $0\le\lambda_i(\omega)\le\bar{\Lambda}$ for every $\omega$, let $p_i\ge0$ be a natural number and $\kappa_i\ge0$ a real, and set $V_i:=Y^{a_i}_{v_i+\lambda_i}-Y^{a_i}_{v_i}$, a random variable with values in the natural numbers by claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process. Then for every $\mathbb{F}_t$-measurable $Z:\Omega\to[0,\infty]$, in $[0,\infty]$ and with the finite product notation: $\mathbb{E}\Bigl[Z\,\prod_{i=1}^{k}\bigl(V_i^{p_i}\,\exp(\kappa_i V_i)\bigr)\Bigr]\ =\ \mathbb{E}\Bigl[Z\,\prod_{i=1}^{k}\mu_{p_i,\kappa_i}(\lambda_i)\Bigr].$

3. (Moment functions) For every real $\kappa\ge0$ and every $x\ge0$: $\mu_{0,\kappa}(x)=\exp\bigl(x\,(\exp(\kappa)-1)\bigr)$, and $\mu_{p,\kappa}(x)$ is finite for every natural number $p\ge0$. For every natural number $p\ge1$ and every $x\ge0$, $\mu_{p,0}(x)=\mu_p(x)$; and for every natural number $p\ge1$, every real $\kappa\ge0$, every real $\bar{\Lambda}\ge0$, and every $x\in[0,\bar{\Lambda}]$, $\mu_{p,\kappa}(x)\ \le\ c_{p,\kappa}(\bar{\Lambda})\,x,\qquad c_{p,\kappa}(\bar{\Lambda})\ :=\ \sum_{k\ge1}\frac{k^{p}\,\exp(\kappa k)\,\bar{\Lambda}^{k-1}}{k!}\ <\ \infty.$

4. (Multi-base window bound) Let $n\ge1$ be a natural number, let $0\le t_1\le\dots\le t_n\le T$, let $Z:\Omega\to[0,\infty]$ be $\mathbb{F}_{t_1}$-measurable, and let $\bar{V},\bar{\Lambda}\ge0$ be reals. Let $\mathcal{A}$ be a finite, possibly empty, set of distinct clock labels, and assign to each $a\in\mathcal{A}$: a base index $b(a)\in\{1,\dots,n\}$; an $\mathbb{F}_{t_{b(a)}}$-measurable anchor $v_a$ with $A^{\vee,a}_{t_{b(a)}}(\omega)\le v_a(\omega)\le\bar{V}$ for every $\omega$; an $\mathbb{F}_{t_{b(a)}}$-measurable width $\lambda_a$ and a real $\bar{\lambda}_a\in[0,\bar{\Lambda}]$ with $0\le\lambda_a(\omega)\le\bar{\lambda}_a$ for every $\omega$; a natural number $p_a\ge0$; and a real $\kappa_a\ge0$. Set $V_a:=Y^{a}_{v_a+\lambda_a}-Y^{a}_{v_a}$, a random variable with values in the natural numbers by claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process. Then, in $[0,\infty]$: $\mathbb{E}\Bigl[Z\,\prod_{a\in\mathcal{A}}\bigl(V_a^{p_a}\,\exp(\kappa_a V_a)\bigr)\Bigr]\ \le\ \Bigl(\prod_{a\in\mathcal{A}}\mathfrak{c}_a\Bigr)\,\mathbb{E}[Z],$ where $\mathfrak{c}_a:=2^{(p_a+1)n}\exp\bigl(n\,\bar{\lambda}_a(\exp(\kappa_a)-1)\bigr)$ if $p_a=0$, and $\mathfrak{c}_a:=2^{(p_a+1)n}\exp\bigl(n\,\bar{\lambda}_a(\exp(\kappa_a)-1)\bigr)\,c_{p_a,\kappa_a}(\bar{\Lambda})\,\bar{\lambda}_a$ if $p_a\ge1$.