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

Preliminaries. (P1) For every clock label $a$, every $j$, and $0\le t\le t'\le T$: $0\le A_j^{a}(t')-A_j^{a}(t)\le B_a(t'-t)$. Indeed, by condition 2 of Solution of the Controlled N-Agent Dynamics the consumed time $A_j^{a}(t)$ is the Lebesgue integral over the compact interval $[0,t]$ of an integrand with values in $[0,B_a]$, and the increment over $(t,t']$ is the integral of that integrand over the complementary piece by additivity of the integral over disjoint measurable sets, hence lies in $[0,B_a(t'-t)]$ by monotonicity. In particular $A_j^{a}(t)\le B_at\le B_aT$, and the frontier $A^{\vee,a}$ is nondecreasing with $A^{\vee,a}_{t'}\le A^{\vee,a}_t+B_a(t'-t)$, the bound passing through the finite maximum. (P2) Each $A_j^{a}(t)$ is $\mathcal{F}^{\mathrm{sys},j}_t$-measurable by part (iv) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, so $A^{\vee,a}_t$ is $\mathbb{F}_t$-measurable; evaluations of the clock paths at random levels are random variables by claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process. Also $\mathbb{F}_s\subseteq\mathbb{F}_w$ for $s\le w$, each $(\mathcal{F}^{\mathrm{sys},j}_t)_t$ being a filtration. (P3) It suffices to prove the claims for bounded $Z$: for $Z:\Omega\to[0,\infty]$ apply the bounded case to $Z\wedge n$ and let $n\to\infty$ by the Monotone Convergence Theorem.

Claim 1. Fix $t$, the label $a$, $p$, and $Z$ with $0\le Z\le\zeta$; fix a natural $m$ and put $\delta=2^{-m}$, $\theta_q=q\delta$. Let $Q$ be the least natural with $\theta_Q\ge\bar{V}$ and partition $\Omega$ into $F_0=\{v=0\}$ and $F_q=\{\theta_{q-1}<v\le\theta_q\}$ for $q=1,\dots,Q$ (using $0\le v\le\bar{V}$). Fix $q$ and write $\theta$ for $\theta_q$, with $\theta=0$ when $q=0$.

Apply Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics at time $r:=t$ with caps $c_a:=\theta$ for the label $a$ and $c_b:=B_bt$ for every label $b\neq a$. By (P1) the constraints for the labels $b\neq a$ hold surely, so the event $C$ there equals $\bigcap_j\{A_j^{a}(t)\le\theta\}$, and $F_q\subseteq C$ because $A_j^{a}(t)\le A^{\vee,a}_t\le v\le\theta$ on $F_q$. Its claim 1 gives $\hat{C}\in\mathcal{H}_q$ (the capped $\sigma$-algebra there) with $P(C\triangle\hat{C})=0$, and its claim 2, applied to the $\mathbb{F}_t$-measurable nonnegative maps $W_q:=Z\mathbf{1}_{F_q}$, $v$, and $\lambda$, gives $\mathcal{H}_q$-measurable $\hat{W}_q,\hat{v}_q,\hat{\lambda}_q$ agreeing with them almost surely on $C$. Set $\lambda_q:=(v+\lambda-\theta)^{+}$ and $\lambda^{\dagger}_q:=\bigl(((\hat{v}_q\wedge\bar{V})+(\hat{\lambda}_q\wedge\bar{\Lambda})-\theta)^{+}\bigr)\wedge\bar{\Lambda}$, an $\mathcal{H}_q$-measurable variable with values in $[0,\bar{\Lambda}]$.

Pathwise split. On $F_q$, with $I:=Y^{a}_{v+\lambda}-Y^{a}_{v}$, $D_q:=Y^{a}_{(v+\lambda)\wedge\theta}-Y^{a}_{v}$, and $G_q:=Y^{a}_{\theta+\lambda_q}-Y^{a}_{\theta}$: if $v+\lambda\ge\theta$ then $(v+\lambda)\wedge\theta=\theta$, $\theta+\lambda_q=v+\lambda$, and $I=D_q+G_q$ by concatenation at the level $\theta$; if $v+\lambda<\theta$ then $\lambda_q=0$, $G_q=0$, $D_q=I$. In all cases $I=D_q+G_q$ with $D_q,G_q\ge0$.

Central term. First, $(\hat{W}_q\wedge\zeta)\mathbf{1}_{\hat{C}}=W_q$ almost surely: the two agree almost surely on $C$ (where $\hat{W}_q=W_q\le\zeta$ and $\mathbf{1}_{\hat{C}}=1$ almost surely), and almost surely off $C$ both vanish ($W_q$ vanishes off $F_q$ and $F_q\subseteq C$, so $W_q=0$ off $C$; and $\mathbf{1}_{\hat{C}}=0$ almost surely off $C$). Second, $\lambda^{\dagger}_q=\lambda_q$ almost surely on $F_q\supseteq\{W_q>0\}$: almost surely on $F_q\subseteq C$ one has $\hat{v}_q=v\le\bar{V}$ and $\hat{\lambda}_q=\lambda\le\bar{\Lambda}$, so the inner truncations do not act, and on $F_q$, $v\le\theta$ gives $\lambda_q\le\lambda\le\bar{\Lambda}$, so the outer truncation does not act either. Hence the integrands of $\mathbb{E}[W_qG_q^{\,p}]$ and $\mathbb{E}\bigl[(\hat{W}_q\wedge\zeta)\mathbf{1}_{\hat{C}}\,(Y^{a}_{\theta+\lambda^{\dagger}_q}-Y^{a}_{\theta})^{p}\bigr]$ agree almost surely, and the expectations coincide. Now apply claim 2 of Predictable-Window Moment Identities for the Homogeneous Poisson Process to the clock $Y^{a}$ with the data: level $\theta$, bounds $\bar{V}':=\theta$ and $\bar{\Lambda}$; $\sigma$-algebra $\mathcal{G}:=\mathcal{H}_q$, which is independent of $\sigma(Y^{a}_{\theta+s}-Y^{a}_{\theta}:s\ge0)$ by claim 3 of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics; window bottom the constant $\theta$; window length $\lambda^{\dagger}_q$; multiplier $(\hat{W}_q\wedge\zeta)\mathbf{1}_{\hat{C}}$. This yields $\mathbb{E}[W_qG_q^{\,p}]=\mathbb{E}\bigl[(\hat{W}_q\wedge\zeta)\mathbf{1}_{\hat{C}}\,\mu_p(\lambda^{\dagger}_q)\bigr]=\mathbb{E}[W_q\,\mu_p(\lambda_q)]$: in the last step the first factors agree almost surely, and where they are positive (a subset of $F_q$ up to a null event) $\lambda^{\dagger}_q=\lambda_q$ almost surely, so the integrands agree almost surely.

Error term. For reals $x\ge y\ge0$, $x^p-y^p=(x-y)(x^{p-1}+\dots+y^{p-1})\le(x-y)\,p\,x^{p-1}$; with $x=I$, $y=G_q$, and $I\le\bar{Y}:=Y^{a}_{\bar{V}+\bar{\Lambda}+1}$ (path monotonicity, $v+\lambda\le\bar{V}+\bar{\Lambda}$), this gives $0\le\mathbb{E}[W_qI^{p}]-\mathbb{E}[W_qG_q^{\,p}]\le p\,\zeta\,\mathbb{E}[\mathbf{1}_{F_q}D_q\bar{Y}^{\,p-1}]$.

Assembly and limit. Summing over $q=0,\dots,Q$ (the $F_q$ partition $\Omega$) and writing $\theta_{(m)}(\omega)$ for the cell top $\delta\lceil v/\delta\rceil$, $\lambda_{(m)}:=(v+\lambda-\theta_{(m)})^{+}$, and $D_{(m)}:=Y^{a}_{(v+\lambda)\wedge\theta_{(m)}}-Y^{a}_{v}$: $\bigl|\mathbb{E}[Z\,I^{p}]-\mathbb{E}[Z\,\mu_p(\lambda_{(m)})]\bigr|\le p\,\zeta\,\mathbb{E}[D_{(m)}\bar{Y}^{\,p-1}].$ As $m\to\infty$: $\theta_{(m)}\downarrow v$ (dyadic upper approximations), so $\lambda_{(m)}\to\lambda$ pointwise and, by continuity and monotonicity of $\mu_p$ (claim 4 of Predictable-Window Moment Identities for the Homogeneous Poisson Process) and the Dominated Convergence Theorem with the constant dominator $\zeta\mu_p(\bar{\Lambda})$, $\mathbb{E}[Z\mu_p(\lambda_{(m)})]\to\mathbb{E}[Z\mu_p(\lambda)]$. Also $D_{(m)}\downarrow0$ pointwise: the levels $(v+\lambda)\wedge\theta_{(m)}$ decrease to $v$, and $Y^{a}$ at levels decreasing to $v$ decreases to $Y^{a}_{v}$ by right-continuity (property 3 of Counting Path and Its Jump Times, as in claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process); with $D_{(m)}\bar{Y}^{\,p-1}\le\bar{Y}^{\,p}$ and $\mathbb{E}[\bar{Y}^{\,p}]=\mu_p(\bar{V}+\bar{\Lambda}+1)<\infty$, dominated convergence gives $\mathbb{E}[D_{(m)}\bar{Y}^{\,p-1}]\to0$. Since $\mathbb{E}[ZI^{p}]$ does not depend on $m$, $\mathbb{E}[Z\,I^{p}]=\mathbb{E}[Z\,\mu_p(\lambda)]$.

The window-event identity is proved by the same split, using $|\mathbf{1}\{I\ge1\}-\mathbf{1}\{G_q\ge1\}|\le D_q$ pointwise (if $I\ge1$ and $G_q=0$ then $D_q=I\ge1$, and $G_q\le I$), the central evaluation $\mathbb{E}[W_q\mathbf{1}\{G_q\ge1\}]=\mathbb{E}[W_q(1-\exp(-\lambda_q))]$ from claim 3 of Predictable-Window Moment Identities for the Homogeneous Poisson Process with the same almost-sure-equality bookkeeping as in the central term, and, in the limit, continuity of $\exp$ with dominator $\zeta$ for the central part together with $\mathbb{E}[Z\,D_{(m)}]\to0$ for the error part, by the same dominated-convergence argument with dominator $\bar{Y}$ and $\mathbb{E}[\bar{Y}]=\mu_1(\bar{V}+\bar{\Lambda}+1)<\infty$. The final inequality follows from the pointwise bound $1-\exp(-\lambda)\le\lambda$: for $\lambda\ge1$ it is trivial, and for $0\le\lambda<1$ the defining series and the geometric partial-sum bound give $\exp(\lambda)\le\sum_k\lambda^k\le1/(1-\lambda)$, so $\exp(-\lambda)\ge1-\lambda$. The case $p=1$ reads $\mathbb{E}[Z(Y^{a}_{v+\lambda}-Y^{a}_{v})]=\mathbb{E}[Z\lambda]$ since $\mu_1(\lambda)=\lambda$.

Claim 2. Assume $Z$ bounded (P3). Write $A^{\vee}:=A^{\vee,a}$ and $\lambda^{*}:=A^{\vee}_s+B_a(w-s)-A^{\vee}_w$; by (P1), $0\le\lambda^{*}\le B_a(w-s)$ and $A^{\vee}_w+\lambda^{*}=A^{\vee}_s+B_a(w-s)$. Since $A^{\vee}_s\le A^{\vee}_w\le A^{\vee}_s+B_a(w-s)$, the increments concatenate pathwise: $Y^{a}_{A^{\vee}_w}-Y^{a}_{A^{\vee}_s}=\bigl[Y^{a}_{A^{\vee}_s+B_a(w-s)}-Y^{a}_{A^{\vee}_s}\bigr]-\bigl[Y^{a}_{A^{\vee}_w+\lambda^{*}}-Y^{a}_{A^{\vee}_w}\bigr],$ both brackets being nonnegative integers bounded by $Y^{a}_{B_aT+B_a(w-s)}$, which is integrable with expectation $\mu_1(B_aT+B_a(w-s))<\infty$. Apply claim 1 with $p=1$ twice: at time $s$, with window bottom $v:=A^{\vee}_s$ ($\mathbb{F}_s$-measurable, $\ge A^{\vee,a}_s$, $\le B_aT=:\bar{V}$), constant window length $B_a(w-s)$, and multiplier $Z$, giving $\mathbb{E}\bigl[Z\bigl(Y^{a}_{A^{\vee}_s+B_a(w-s)}-Y^{a}_{A^{\vee}_s}\bigr)\bigr]=B_a(w-s)\,\mathbb{E}[Z]$; and at time $w$, with window bottom $v:=A^{\vee}_w$ ($\mathbb{F}_w$-measurable, $\ge A^{\vee,a}_w$, $\le\bar{V}$), window length $\lambda^{*}$ ($\mathbb{F}_w$-measurable, values in $[0,B_a(w-s)]$), and multiplier $Z$ (which is $\mathbb{F}_w$-measurable by (P2)), giving $\mathbb{E}\bigl[Z\bigl(Y^{a}_{A^{\vee}_w+\lambda^{*}}-Y^{a}_{A^{\vee}_w}\bigr)\bigr]=\mathbb{E}[Z\lambda^{*}]$. Subtracting (all terms finite), $\mathbb{E}\bigl[Z\bigl(Y^{a}_{A^{\vee}_w}-Y^{a}_{A^{\vee}_s}\bigr)\bigr]=\mathbb{E}\bigl[Z\bigl(B_a(w-s)-\lambda^{*}\bigr)\bigr]=\mathbb{E}\bigl[Z\bigl(A^{\vee}_w-A^{\vee}_s\bigr)\bigr].$

Claim 3. By induction on $k$; the case $k=1$ is claim 1, in both the power and the indicator form. Let $k\ge2$ and suppose the claim holds for every choice of $k-1$ distinct labels. Fix the data of the claim, abbreviate $I_n:=Y^{a_n}_{v_n+\lambda_n}-Y^{a_n}_{v_n}$, and for each $n$ let $f_n$ denote either $x\mapsto x^{p_n}$ or $x\mapsto\mathbf{1}\{x\ge1\}$, with $g_n(\lambda)$ the corresponding value $\mu_{p_n}(\lambda)$ or $1-\exp(-\lambda)$. By (P3) assume $0\le Z\le\zeta$.

Run the cell decomposition of the claim 1 proof for the label $a:=a_k$ and the window bottom $v:=v_k$: for fixed natural $m$, cells $F_0,\dots,F_Q$ with tops $\theta:=\theta_q$, and, for each $q$, the application of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics at time $t$ with caps $c_{a_k}:=\theta$ and $c_b:=B_bt$ for $b\neq a_k$, giving $C\supseteq F_q$, $\hat{C}\in\mathcal{H}_q$, and $\mathcal{H}_q$-measurable versions $\hat{W}_q$ of $W_q:=Z\mathbf{1}_{F_q}$ and $\hat{v}_n,\hat{\lambda}_n$ of $v_n,\lambda_n$ for all $n\le k$, agreeing with the unhatted maps almost surely on $C$; and the quantities $\lambda_q$, $\lambda^{\dagger}_q$, $D_q$, $G_q$ built from $v_k,\lambda_k,\hat{v}_k,\hat{\lambda}_k$ exactly as there.

Let $\mathcal{K}_q:=\sigma\bigl(\mathcal{H}_q\cup\bigcup_{n<k}\sigma(Y^{a_n}_u:u\ge0)\bigr)$. Every generator of $\mathcal{K}_q$ lies in the $\sigma$-algebra $\mathcal{H}^{+}_{a_k}$ of claim 3 of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics (for the caps above): the generators of $\mathcal{H}_q$ do, as shown there, and the full variable families of the clocks $a_n$, $n<k$, do since $a_n\neq a_k$. Hence $\sigma(Y^{a_k}_{\theta+s}-Y^{a_k}_{\theta}:s\ge0)$ is independent of $\mathcal{K}_q$. Define, for $n<k$, the counts $\hat{I}_n:=Y^{a_n}_{(\hat{v}_n\wedge\bar{V})+(\hat{\lambda}_n\wedge\bar{\Lambda})}-Y^{a_n}_{\hat{v}_n\wedge\bar{V}}$; these are $\mathcal{K}_q$-measurable by the final clause of claim 1 of Predictable-Window Moment Identities for the Homogeneous Poisson Process, applied with $\mathcal{G}_0:=\mathcal{K}_q$ (the evaluation levels are $\mathcal{H}_q$-measurable and every variable $Y^{a_n}_u$ is $\mathcal{K}_q$-measurable); almost surely on $C$ the truncations do not act and $\hat{I}_n=I_n$.

Apply claim 2 (respectively claim 3, for the indicator form of $f_k$) of Predictable-Window Moment Identities for the Homogeneous Poisson Process to the clock $Y^{a_k}$ with level $\theta$, $\sigma$-algebra $\mathcal{G}:=\mathcal{K}_q$, window bottom the constant $\theta$, window length $\lambda^{\dagger}_q$, and the $\mathcal{K}_q$-measurable $[0,\infty]$-valued multiplier $(\hat{W}_q\wedge\zeta)\,\mathbf{1}_{\hat{C}}\,\prod_{n<k}f_n(\hat{I}_n)$. By the almost-sure-equality bookkeeping of the claim 1 proof (extended by $\hat{I}_n=I_n$ almost surely on $C$, both sides vanishing almost surely off $C$), $\mathbb{E}\Bigl[W_q\,f_k(G_q)\prod_{n<k}f_n(I_n)\Bigr]=\mathbb{E}\Bigl[W_q\,g_k(\lambda_q)\prod_{n<k}f_n(I_n)\Bigr].$ The error of replacing $f_k(I_k)$ by $f_k(G_q)$ is bounded as in the claim 1 proof, with every estimate multiplied by $\prod_{n<k}f_n(I_n)\le\prod_{n<k}\bar{Y}_n^{\,p_n}$, where $\bar{Y}_n:=Y^{a_n}_{\bar{V}+\bar{\Lambda}+1}$: in the power case by $p_k\zeta\,\mathbb{E}[\mathbf{1}_{F_q}D_q\bar{Y}_k^{\,p_k-1}\prod_{n<k}\bar{Y}_n^{\,p_n}]$, in the indicator case by $\zeta\,\mathbb{E}[\mathbf{1}_{F_q}D_q\prod_{n<k}\bar{Y}_n^{\,p_n}]$. The dominating product $\bar{Y}_k^{\,p_k}\prod_{n<k}\bar{Y}_n^{\,p_n}$ is integrable with expectation $\prod_{n\le k}\mu_{p_n}(\bar{V}+\bar{\Lambda}+1)<\infty$: the variables $\bar{Y}_1,\dots,\bar{Y}_k$ are functions of the variables of $k$ distinct clocks, whose $\sigma$-algebras form an independent family (hypothesis of Level-Revealed Conditioning for Jointly Driven Solutions of the Controlled N-Agent Dynamics), so each partial product $\prod_{n\le j}\bar{Y}_n^{\,p_n}$ is independent of $\bar{Y}_{j+1}^{\,p_{j+1}}$ by part (a) of Grouping Lemma for Independent Random Variables together with Joint Distribution, Expectations, and Block Independence for Independent Random Variables --- functions of disjoint blocks of an independent family are independent --- and Expectation of a Product of Independent Random Variables applies inductively, each factor having finite expectation $\mu_{p_n}(\bar{V}+\bar{\Lambda}+1)$ by claim 4 of Predictable-Window Moment Identities for the Homogeneous Poisson Process and claim 2 of Image Measures, Measures with Densities, and Change of Variables. Summing over $q$ and letting $m\to\infty$ exactly as in the claim 1 proof --- the same pointwise limits $\lambda_{(m)}\to\lambda_k$ and $D_{(m)}\downarrow0$, with dominated convergence now using the integrable dominators $\zeta\,\mu_{p_k}(\bar{\Lambda})\prod_{n<k}\bar{Y}_n^{\,p_n}$ in the power case, $\zeta\prod_{n<k}\bar{Y}_n^{\,p_n}$ in the indicator case, and $\bar{Y}_k^{\,p_k}\prod_{n<k}\bar{Y}_n^{\,p_n}$ for the error terms --- yields $\mathbb{E}\Bigl[Z\,\prod_{n=1}^{k}f_n(I_n)\Bigr]=\mathbb{E}\Bigl[Z\,g_k(\lambda_k)\prod_{n=1}^{k-1}f_n(I_n)\Bigr].$ The multiplier $Z\,g_k(\lambda_k)$ is $\mathbb{F}_t$-measurable and nonnegative ($\lambda_k$ is $\mathbb{F}_t$-measurable and $g_k$ is Borel), so the induction hypothesis for the labels $a_1,\dots,a_{k-1}$ applies and gives $\mathbb{E}[Z\prod_{n\le k}f_n(I_n)]=\mathbb{E}[Z\prod_{n\le k}g_n(\lambda_n)]$, completing the induction; the monotone reduction of (P3) removes the boundedness of $Z$.