Proof of Below-Frontier Evaluations, Tilted Poisson Moment Functions, and Multi-Base Window Bounds for Jointly Driven Solutions of the Controlled N-Agent Dynamics

We record the standing facts, all valid at every point of $\Omega$. Each clock $Y^{a}$ is a homogeneous Poisson process with rate $1$ whose paths are counting paths, by conditions 2 and 3 of N-Agent Driving System; in particular every path is nondecreasing, right-continuous, integer-valued, and finite at every level. Evaluations of the clock paths at nonnegative random levels, and increments of a clock between two ordered random levels, are random variables with values in the natural numbers, by claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process; this applies to every window count below. By condition 2 of Solution of the Controlled N-Agent Dynamics, the consumed clock times $A_j^{a}$ of each of the $J$ given solutions are defined on all of $\Omega$ with $0\le A_j^{a}(t)\le B_at$, vanish identically off the regular event, and at points of the regular event are integrals over time intervals of integrands bounded by $B_a$; there, splitting the interval of integration and using linearity and monotonicity of the integral, the increments satisfy $0\le A_j^{a}(t')-A_j^{a}(t)\le B_a(t'-t)$ for $t\le t'$, while off the regular event the paths vanish; so every path of $A_j^{a}$ is nondecreasing and continuous. The counters $N_j^{a}(t)=Y^{a}_{A_j^{a}(t)}$, the evaluations of the clock paths at the consumed levels, as well as the consumed times themselves, are adapted to the $j$-th system filtration by the adaptedness assertion of part (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, hence $\mathbb{F}_t$-measurable at each $t$; and $\mathbb{F}_s\subseteq\mathbb{F}_t$ for $s\le t$, each system filtration being a filtration, which is used below whenever a variable measurable at an earlier base time is employed at a later one. Consequently $A^{\vee,a}$ has continuous nondecreasing paths with $A^{\vee,a}_0=0$ and $A^{\vee,a}_t\le B_aT$, each $A^{\vee,a}_t$ is $\mathbb{F}_t$-measurable, and, the clock path being nondecreasing, the frontier count $\mathcal{Y}^{\vee,a}_t:=Y^{a}_{A^{\vee,a}_t}=\max(N_1^{a}(t),\dots,N_J^{a}(t))$ is $\mathbb{F}_t$-measurable with right-continuous nondecreasing paths, a right-continuous path composed with a continuous nondecreasing one.

Claim 1. Define $\tau:=\inf\{s\in[0,t]:A^{\vee,a}_s\ge W\}$. The set is nonempty, containing $t$; by continuity of the path $s\mapsto A^{\vee,a}_s$ it is closed, so the infimum is a minimum and $A^{\vee,a}_{\tau}\ge W$; and for $s<\tau$ one has $A^{\vee,a}_s<W$, so by continuity from the left $A^{\vee,a}_{\tau}\le W$ when $\tau>0$, while for $\tau=0$ the value $A^{\vee,a}_0=0\le W$ gives the same; hence $A^{\vee,a}_{\tau}=W$ and $Y^{a}_{W}=Y^{a}_{A^{\vee,a}_{\tau}}=\mathcal{Y}^{\vee,a}_{\tau}$. For every $s\in[0,t]$, $\{\tau\le s\}=\{A^{\vee,a}_s\ge W\}$ (if $A^{\vee,a}_s\ge W$ then $s$ lies in the set, and conversely $\tau\le s$ gives $A^{\vee,a}_s\ge A^{\vee,a}_{\tau}\ge W$ by monotonicity), a member of $\mathbb{F}_t$, both entries being $\mathbb{F}_t$-measurable. Let $\tau_i:=\min\bigl(t,\,2^{-i}\lceil2^{i}\tau\rceil\bigr)$ for natural $i\ge1$: each $\tau_i$ takes finitely many values in the dyadic grid of $[0,t]$ together with $t$, each value-set $\{\tau_i=q\}$ lies in $\mathbb{F}_t$, and $\tau_i\downarrow\tau$ with $\tau_i\ge\tau$ (for $\tau=t$ every $\tau_i=t$). Hence $\mathcal{Y}^{\vee,a}_{\tau_i}=\sum_{q}\mathbf{1}\{\tau_i=q\}\,\mathcal{Y}^{\vee,a}_{q}$ is $\mathbb{F}_t$-measurable, the values $q$ being at most $t$ and the filtrations nested, and by right-continuity of the frontier-count path $\mathcal{Y}^{\vee,a}_{\tau_i}\to\mathcal{Y}^{\vee,a}_{\tau}$ pointwise; the values being natural numbers, the sequence is eventually constant at every $\omega$, so for every real $c$ the set where $Y^{a}_{W}\ge c$ equals $\bigcup_{i_0\ge1}\bigcap_{i\ge i_0}\{\mathcal{Y}^{\vee,a}_{\tau_i}\ge c\}$, a member of $\mathbb{F}_t$; thus $Y^{a}_{W}=\mathcal{Y}^{\vee,a}_{\tau}$ is $\mathbb{F}_t$-measurable.

Claim 2. Every path of $Y^{a_i}$ is finite at every level, so each $V_i$ is a finite natural number everywhere, and by the series form of the real exponential function, pointwise, $V_i^{p_i}\exp(\kappa_iV_i)\ =\ \sum_{j\ge0}\frac{\kappa_i^{j}}{j!}\,V_i^{p_i+j},$ a series of nonnegative terms. Multiplying the $k$ series and expanding, the product $\prod_{i=1}^{k}\bigl(V_i^{p_i}\exp(\kappa_iV_i)\bigr)$ is the least upper bound of the partial sums of the countable family indexed by $(j_1,\dots,j_k)$ with terms $\prod_{i}(\kappa_i^{j_i}/j_i!)\,V_i^{p_i+j_i}$, by the Tonelli theorem applied to the counting measures on the index sets. By the monotone convergence theorem applied along an exhausting nondecreasing sequence of finite index sets, and by linearity of the integral of nonnegative functions over each finite partial sum, $\mathbb{E}\Bigl[Z\prod_{i=1}^{k}\bigl(V_i^{p_i}\exp(\kappa_iV_i)\bigr)\Bigr]\ =\ \sum_{(j_1,\dots,j_k)}\ \prod_{i}\frac{\kappa_i^{j_i}}{j_i!}\ \mathbb{E}\Bigl[Z\prod_{i}V_i^{p_i+j_i}\Bigr].$ For a fixed index $(j_1,\dots,j_k)$: dropping the factors with $p_i+j_i=0$, which equal $1$ (the convention $0^{0}=1$ of the statement), claim 3 of Frontier-Window and Crossing-Compensation Identities for Jointly Driven Solutions of the Controlled N-Agent Dynamics applied at the base time $t$ to the labels with $p_i+j_i\ge1$ --- a product-window identity with the multiplier $Z$, the anchors $v_i$, and the widths $\lambda_i$ --- gives $\mathbb{E}[Z\prod_iV_i^{p_i+j_i}]=\mathbb{E}[Z\prod_i\mu_{p_i+j_i}(\lambda_i)]$, with the convention $\mu_0:=1$ for the dropped factors, and with the value $\mathbb{E}[Z]$ when every exponent vanishes. Summing back, again by monotone convergence and linearity, $\mathbb{E}\Bigl[Z\prod_{i=1}^{k}\bigl(V_i^{p_i}\exp(\kappa_iV_i)\bigr)\Bigr]\ =\ \mathbb{E}\Bigl[Z\,\sum_{(j_1,\dots,j_k)}\prod_{i}\Bigl(\frac{\kappa_i^{j_i}}{j_i!}\,\mu_{p_i+j_i}(\lambda_i)\Bigr)\Bigr]\ =\ \mathbb{E}\Bigl[Z\prod_{i=1}^{k}\Bigl(\sum_{j\ge0}\frac{\kappa_i^{j}}{j!}\,\mu_{p_i+j}(\lambda_i)\Bigr)\Bigr],$ the multi-indexed sum of the products factorizing into the product of the single-index sums by the Tonelli theorem once more. It remains to identify, for every natural number $p\ge0$, real $\kappa\ge0$, and $x\ge0$, $\sum_{j\ge0}\frac{\kappa^{j}}{j!}\,\mu_{p+j}(x)\ =\ \mu_{p,\kappa}(x).$ By claim 4 of Predictable-Window Moment Identities for the Homogeneous Poisson Process, $\mu_{q}(x)=\exp(-x)\sum_{k\ge0}k^{q}x^{k}/k!$ for $q\ge1$, the term $k=0$ vanishing; and the same expression at $q=0$ evaluates, with the convention $0^{0}=1$, to $\exp(-x)\exp(x)=1=\mu_0$, by the series form of the exponential and its functional equation. Interchanging the two summations of nonnegative terms (the Tonelli theorem again), $\sum_{j\ge0}\frac{\kappa^{j}}{j!}\exp(-x)\sum_{k\ge0}\frac{k^{p+j}x^{k}}{k!}\ =\ \exp(-x)\sum_{k\ge0}\frac{k^{p}x^{k}}{k!}\sum_{j\ge0}\frac{(\kappa k)^{j}}{j!}\ =\ \exp(-x)\sum_{k\ge0}\frac{k^{p}\exp(\kappa k)x^{k}}{k!}\ =\ \mu_{p,\kappa}(x).$ This proves claim 2.

Claim 3. The identity $\mu_{p,0}=\mu_p$ for $p\ge1$ is the series form just quoted. For $p=0$: $\mu_{0,\kappa}(x)=\exp(-x)\sum_{k\ge0}(x\exp(\kappa))^{k}/k!=\exp(-x)\exp(x\exp(\kappa))=\exp(x(\exp(\kappa)-1))$, by the series form of the exponential and its functional equation. For $p\ge1$ and $0\le x\le\bar{\Lambda}$: since $\exp(-x)\le1$ (for $x\ge0$ the series gives $\exp(x)\ge1$, and $\exp(-x)\exp(x)=1$ by the functional equation, so $\exp(-x)\le1$), $\mu_{p,\kappa}(x)\ \le\ \sum_{k\ge1}\frac{k^{p}\exp(\kappa k)x^{k}}{k!}\ =\ x\,\sum_{k\ge1}\frac{k^{p}\exp(\kappa k)x^{k-1}}{k!}\ \le\ x\,c_{p,\kappa}(\bar{\Lambda}).$ Finiteness of $c_{p,\kappa}(\bar{\Lambda})$: for $k\ge2p$ each of the $p$ factors of $k(k-1)\cdots(k-p+1)$ is at least $k/2$, so $k^{p}\le2^{p}\,k(k-1)\cdots(k-p+1)$ and, reindexing by $r=k-p$, $\sum_{k\ge2p}\frac{k^{p}\exp(\kappa k)\bar{\Lambda}^{k-1}}{k!}\ \le\ 2^{p}\exp(\kappa p)\sum_{r\ge p}\frac{\exp(\kappa r)\bar{\Lambda}^{r+p-1}}{r!}\ \le\ 2^{p}\exp(\kappa p)\,\max(1,\bar{\Lambda}^{p-1})\,\exp\bigl(\bar{\Lambda}\exp(\kappa)\bigr),$ using $\bar\Lambda^{r+p-1}\le\max(1,\bar\Lambda^{p-1})\bar\Lambda^{r}$ and the series of the exponential once more; the finitely many terms with $k<2p$ are finite. Finiteness of $\mu_{p,\kappa}(x)$ for $p\ge1$ follows from the linear bound with $\bar{\Lambda}:=x$; for $p=0$ it is immediate from the closed form just established.

Claim 4. We induct on $n$. Throughout, for a label $a\in\mathcal{A}$, write $\Xi_a:=\bar{\lambda}_a(\exp(\kappa_a)-1)$, $e_a:=\exp(\Xi_a)$, and, when $p_a\ge1$, $\gamma_a:=c_{p_a,\kappa_a}(\bar{\Lambda})\,\bar{\lambda}_a$; set $\delta_a:=\gamma_a$ if $p_a\ge1$ and $\delta_a:=e_a$ if $p_a=0$, so that $\mathfrak{c}_a=2^{(p_a+1)n}\exp(n\Xi_a)$ for $p_a=0$ and $=2^{(p_a+1)n}\exp(n\Xi_a)\gamma_a$ for $p_a\ge1$. By claim 3 and the monotonicity of the exponential, pathwise $\mu_{p_a,\kappa_a}(\lambda_a)\le\gamma_a$ for $p_a\ge1$ and $\mu_{0,\kappa_a}(\lambda_a)=\exp(\lambda_a(\exp(\kappa_a)-1))\le e_a$; in either case $\mu_{p_a,\kappa_a}(\lambda_a)\le\delta_a$.

Base case $n=1$. All windows share the base $t_1$, and $Z$ is $\mathbb{F}_{t_1}$-measurable; claim 2 and the pathwise bounds just recorded give $\mathbb{E}[Z\prod_a\bigl(V_a^{p_a}\exp(\kappa_aV_a)\bigr)]=\mathbb{E}[Z\prod_a\mu_{p_a,\kappa_a}(\lambda_a)]\le\prod_a\delta_a\,\mathbb{E}[Z]\le\prod_a\mathfrak{c}_a\,\mathbb{E}[Z]$. (For $\mathcal{A}=\varnothing$ the claim is trivial.)

Inductive step. Let $n\ge2$ and assume the claim for $n-1$ (with the same $\bar{V}$, $\bar{\Lambda}$, and caps). If no label is based at index $n$, the claim follows from the induction hypothesis for the chain $t_1\le\dots\le t_{n-1}$, the constants being nondecreasing in $n$. Otherwise write $\mathcal{A}_n:=\{a:b(a)=n\}$ and $\mathcal{A}_-:=\mathcal{A}\setminus\mathcal{A}_n$. For $a\in\mathcal{A}_-$ define $\tilde{v}_a:=\max\bigl(v_a,A^{\vee,a}_{t_n}\bigr),\qquad \tilde{\lambda}_a:=\bigl(v_a+\lambda_a-\tilde{v}_a\bigr)^{+},$ and the split $V^{-}_a:=Y^{a}_{(v_a+\lambda_a)\wedge A^{\vee,a}_{t_n}}-Y^{a}_{v_a\wedge A^{\vee,a}_{t_n}}$ and $\tilde{V}_a:=Y^{a}_{\tilde{v}_a+\tilde{\lambda}_a}-Y^{a}_{\tilde{v}_a}$, both random variables by the standing facts. Then, at every $\omega$, $V_a=V^{-}_a+\tilde{V}_a$: if $A^{\vee,a}_{t_n}\le v_a$ then $V^{-}_a=0$, $\tilde{v}_a=v_a$, $\tilde{\lambda}_a=\lambda_a$, $\tilde{V}_a=V_a$; if $v_a<A^{\vee,a}_{t_n}<v_a+\lambda_a$ then $V^{-}_a$ counts the levels in the part of the window up to $A^{\vee,a}_{t_n}$ and $\tilde{V}_a$ the rest; and if $A^{\vee,a}_{t_n}\ge v_a+\lambda_a$ then $V^{-}_a=V_a$ and $\tilde{\lambda}_a=0$, $\tilde{V}_a=0$. Moreover $V^{-}_a$ is $\mathbb{F}_{t_n}$-measurable by claim 1, both evaluation levels being $\mathbb{F}_{t_n}$-measurable and bounded by $A^{\vee,a}_{t_n}$ everywhere; and $\tilde{V}_a$ is a window on the clock $a$ with base time $t_n$: its anchor $\tilde{v}_a$ is $\mathbb{F}_{t_n}$-measurable with $A^{\vee,a}_{t_n}\le\tilde{v}_a\le\max(\bar{V},B_aT)$ everywhere, and its width $\tilde{\lambda}_a$ is $\mathbb{F}_{t_n}$-measurable with $0\le\tilde{\lambda}_a\le\lambda_a\le\bar{\lambda}_a$.

Pathwise, for $a\in\mathcal{A}_-$ with $p_a\ge1$, using $(x+y)^{p}\le2^{p}(x^{p}+y^{p})$ for nonnegative reals ($x+y\le2\max(x,y)$, and the $p$-th power is nondecreasing on the nonnegative reals) and the exact factorization $\exp(\kappa_aV_a)=\exp(\kappa_aV^{-}_a)\exp(\kappa_a\tilde{V}_a)$, from the functional equation of the exponential: $V_a^{p_a}\exp(\kappa_aV_a)\ \le\ 2^{p_a}\Bigl[\exp(\kappa_aV^{-}_a)\cdot\tilde{V}_a^{p_a}\exp(\kappa_a\tilde{V}_a)\ +\ (V^{-}_a)^{p_a}\exp(\kappa_aV^{-}_a)\cdot\exp(\kappa_a\tilde{V}_a)\Bigr],$ while for $p_a=0$ the factorization is exact with a single term. Expanding the product over $a\in\mathcal{A}_-$ distributively, the left side of the claim is at most the sum, over the subsets $S$ of $\{a\in\mathcal{A}_-:p_a\ge1\}$, of $\prod_{a\in\mathcal{A}_-}2^{p_a}$ times $\mathbb{E}\Bigl[\ \underbrace{Z\prod_{a\in S}\exp(\kappa_aV^{-}_a)\prod_{a\in\mathcal{A}_-\setminus S}\bigl((V^{-}_a)^{p_a}\exp(\kappa_aV^{-}_a)\bigr)}_{\text{the multiplier}}\ \cdot\ \prod_{a\in\mathcal{A}_n}\bigl(V_a^{p_a}\exp(\kappa_aV_a)\bigr)\ \cdot\ \prod_{a\in S}\bigl(\tilde{V}_a^{p_a}\exp(\kappa_a\tilde{V}_a)\bigr)\ \cdot\ \prod_{a\in\mathcal{A}_-\setminus S}\exp(\kappa_a\tilde{V}_a)\ \Bigr],$ with the convention that for $a\in\mathcal{A}_-$ with $p_a=0$ the label lies in $\mathcal{A}_-\setminus S$ always. The multiplier is $\mathbb{F}_{t_n}$-measurable and nonnegative, and the remaining factors are tilted windows at the common base $t_n$ on distinct clocks --- the labels of $\mathcal{A}_n$ with their original data, the labels of $S$ with powers $p_a$ and widths $\tilde{\lambda}_a$, and the labels of $\mathcal{A}_-\setminus S$ with powers $0$ and widths $\tilde{\lambda}_a$ --- all anchors bounded by $\max(\bar{V},\max_aB_aT)$ and all widths by $\bar{\Lambda}$. Claim 2 (with that anchor bound in place of $\bar{V}$) and the pathwise bounds on the $\mu$-values give for this expectation the upper bound $\prod_{a\in\mathcal{A}_n}\delta_a\ \cdot\ \prod_{a\in S}\gamma_a\ \cdot\ \prod_{a\in\mathcal{A}_-\setminus S}e_a\ \cdot\ \mathbb{E}\Bigl[Z\prod_{a\in S}\exp(\kappa_aV^{-}_a)\prod_{a\in\mathcal{A}_-\setminus S}\bigl((V^{-}_a)^{p_a}\exp(\kappa_aV^{-}_a)\bigr)\Bigr],$ using for the widths $\tilde\lambda_a\le\bar\lambda_a$. Since $0\le V^{-}_a\le V_a$ pointwise and $\kappa_a\ge0$, the last expectation is at most $\mathbb{E}[Z\prod_{a\in S}\exp(\kappa_aV_a)\prod_{a\in\mathcal{A}_-\setminus S}\bigl(V_a^{p_a}\exp(\kappa_aV_a)\bigr)]$, which is an instance of the claim for the chain $t_1\le\dots\le t_{n-1}$ with the original windows of $\mathcal{A}_-$, the powers replaced by $0$ on $S$, and the same multiplier $Z$; the induction hypothesis bounds it by $\prod_{a\in S}2^{(0+1)(n-1)}\exp((n-1)\Xi_a)\cdot\prod_{a\in\mathcal{A}_-\setminus S}\mathfrak{c}^{(n-1)}_a\cdot\mathbb{E}[Z]$, with $\mathfrak{c}^{(n-1)}_a$ the claim's constant for $n-1$ bases.

Collecting, and recombining the sum over $S$ into a product over $a\in\mathcal{A}_-$ of the two branch values, the left side of the claim is at most $\prod_{a\in\mathcal{A}_n}\delta_a\cdot\prod_{a\in\mathcal{A}_-}\sigma_a\cdot\mathbb{E}[Z]$, where for $p_a\ge1$ $\sigma_a\ =\ 2^{p_a}\Bigl[\gamma_a\,2^{n-1}\exp((n-1)\Xi_a)\ +\ e_a\,2^{(p_a+1)(n-1)}\exp((n-1)\Xi_a)\,\gamma_a\Bigr]\ \le\ \gamma_a\exp(n\Xi_a)\,2^{p_a}\Bigl[2^{n-1}+2^{(p_a+1)(n-1)}\Bigr]\ \le\ \gamma_a\exp(n\Xi_a)\,2^{p_a+1+(p_a+1)(n-1)}\ =\ \mathfrak{c}_a,$ using $e_a\exp((n-1)\Xi_a)=\exp(n\Xi_a)$, $\exp((n-1)\Xi_a)\le\exp(n\Xi_a)$, $2^{n-1}\le2^{(p_a+1)(n-1)}$, and $p_a+1+(p_a+1)(n-1)=(p_a+1)n$; while for $p_a=0$ there is a single branch and $\sigma_a=e_a\cdot2^{n-1}\exp((n-1)\Xi_a)=2^{n-1}\exp(n\Xi_a)\le\mathfrak{c}_a$. Finally, for $a\in\mathcal{A}_n$ with $p_a\ge1$, $\gamma_a\le2^{(p_a+1)n}\exp(n\Xi_a)\,\gamma_a=\mathfrak{c}_a$, and for $a\in\mathcal{A}_n$ with $p_a=0$, $e_a=\exp(\Xi_a)\le2^{n}\exp(n\Xi_a)=\mathfrak{c}_a$. This closes the induction and proves claim 4.