Proof of Post-Exit Comparison at a Stopping Time for the Recentred N-Agent Cost under To-Go Value Regularity

Throughout, $\lambda_{[a,b]}$ and $\mathcal{B}_{[a,b]}$ are the restricted Lebesgue measure and $\sigma$-algebra on a compact interval, linearity and monotonicity of the integral are used freely (in particular the scalar bound $|\int h|\le\int|h|$, from monotonicity applied to $\pm h\le|h|$), a bounded measurable real-valued function on a compact interval is integrable, and expectations of bounded random variables exist by the same facts on $(\Omega,\mathcal{F},P)$. Write $K'_{M}=1+2(l-1)B\,T$ for the everywhere bound $|M^{\gamma}_{t}|\le K'_{M}$ of claim 2 of the stopped covariation lemma; by the same claim, for $\omega\in\Omega_{0}$ each path $t\mapsto M^{\gamma}_{t}(\omega)$ and $t\mapsto\Sigma^{\gamma}_{t}(\omega)$ is right-continuous at every $t\in[0,T)$ in the $\varepsilon$-$\eta$ sense, and the families $\mathbf{1}_{\Omega_{0}}M^{\gamma}$ and $\mathbf{1}_{\Omega_{0}}\Sigma^{\gamma}$ are progressively measurable; taking the time index $T$ in the definition of progressive measurability, both are also measurable with respect to the product $\sigma$-algebra $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$, and their paths are measurable on $[0,T]$. By part (a) of the first-order expansion lemma, $(t,\omega)\mapsto\mathbf{1}_{\Omega_{0}}\mathcal{D}_{t}$ and $(t,\omega)\mapsto\mathbf{1}_{\Omega_{0}}L(\Sigma_{t},\alpha_{t})$ are product-measurable with $|\mathcal{D}_{t}|\le C_{\mathcal{D}}$ and $|L(\Sigma_{t},\alpha_{t})|\le C_{LG}$ at every point at which $\Sigma_{t}\in\Delta^{l}$ and $\alpha_{t}\in\mathcal{A}$ (hence everywhere, by the solution definition), so their paths at each $\omega\in\Omega_{0}$ are measurable and bounded. For $\gamma\in\{1,\dots,l\}$ put $f^{\gamma}(s)=\partial_{\gamma}\mathcal{H}_{s}(S_{s},A_{s})$; each $f^{\gamma}$ is continuous on $[0,T]$ with $\sum_{\gamma=1}^{l}|f^{\gamma}(s)|\le C_{\partial}$, as recorded in the expansion lemma. Finally, the paths $t\mapsto S^{\gamma}_{t}$, $t\mapsto A^{j}_{t}$ and $t\mapsto P^{\gamma}_{t}$ are continuous (clause 1 of the trajectory-pair and co-state definitions), with $\sum_{\gamma}|P^{\gamma}_{t}|\le C_{P}$; and by clause 1 of the cost extension definition and part (i) of the regularity of the extended drift, $L=\bar{L}$ on $\Delta^{l}\times\mathbb{R}^{m}$, $G=\bar{G}$ on $\Delta^{l}$, and $b^{\delta}=\bar{b}^{\delta}$ on $\Delta^{l}\times\mathcal{A}$, so that at every $(t,\omega)$ with $\Sigma_{t}\in\Delta^{l}$, $\alpha_{t}\in\mathcal{A}$,

$$L(\Sigma_{t},\alpha_{t})-L(S_{t},A_{t})=\mathcal{H}_{t}(\Sigma_{t},\alpha_{t})-\mathcal{H}_{t}(S_{t},A_{t})+\sum_{\delta=1}^{l}P^{\delta}_{t}\,g^{\delta}(t),\qquad \mathcal{H}_{t}(\Sigma_{t},\alpha_{t})-\mathcal{H}_{t}(S_{t},A_{t})=\mathcal{D}_{t}+\sum_{\gamma=1}^{l}f^{\gamma}(t)\,y^{\gamma}_{t},$$

where $g^{\delta}(s)=b^{\delta}(\Sigma_{s},\alpha_{s})-b^{\delta}(S_{s},A_{s})$, the first identity by the definition of $\mathcal{H}$ and the second by the definition of $\mathcal{D}_{t}$ in the expansion lemma.

Step 0: interval identities. Exactly as in the proof of the time-shift lemma we use, for real numbers $a<b$, $c$, and a measurable bounded $\varphi$ on the relevant interval: (0.1) the map $s\mapsto\varphi(s+c)$ on $[a,b]$ is measurable and bounded whenever $\varphi$ is so on $[a+c,b+c]$, with equal integrals — via the zero extensions of claim 2 of the toolkit, which satisfy $\tilde{g}(x)=\tilde{\varphi}(x+c)$ pointwise, the positive and negative parts of the integral definition, and claims 1 and 2 of translation invariance; (0.2) for $a<c<b$, $\int_{[a,b]}\varphi=\int_{[a,c]}\varphi+\int_{[c,b]}\varphi$ (restrictions), via the decomposition of $\varphi$ into the products of $\varphi$ with the indicators of $[a,c]$ and $(c,b]$ — measurable by arithmetic of measurable functions — claim 2 of the toolkit, and claim 2 of the null-set lemma for the single-point discrepancy at $c$; and (0.3) for $r\in[a,b]$ and $\varphi$ measurable bounded on $[a,b]$, $\int_{[0,T]}$-type cut identities $\int_{[a,b]}\mathbf{1}_{\{s\ge r\}}\varphi(s)\,ds=\int_{[r,b]}\varphi$ for $r<b$, and $=0$ for $r=b$ (in the first case by the same indicator decomposition and null-set step; in the second because the integrand vanishes off the null set $\{b\}$, claim 1 of the null-set lemma).

Step 1: forward form of the co-state. For $\gamma\in\{1,\dots,l\}$, the function $-f^{\gamma}$ is the integrand of clause 2 of the co-state definition, and exactly as in Step 2 of the proof of the mean-field expansion lemma — clause 2 of the co-state definition at the two times, claim 3 of the toolkit, and the splitting (0.2) — one obtains

$$P^{\gamma}_{t}=P^{\gamma}_{0}+\int_{[0,t]}f^{\gamma}(s)\,ds\quad(t\in[0,T]),\qquad P^{\gamma}_{T}=-\partial_{\gamma}\bar{G}(S_{T}),\qquad \int_{[r,T]}f^{\gamma}(s)\,ds=P^{\gamma}_{T}-P^{\gamma}_{r}\quad(r\in[0,T)),$$

the last display also holding for $r=T$ with the convention $\int_{[T,T]}=0$.

Step 2: proof of claim 1. The path measurability and boundedness statements were collected above ($y^{\gamma}_{t}=\Sigma^{\gamma}_{t}-S^{\gamma}_{t}$ with $|y^{\gamma}_{t}|\le1$, both terms lying in $[0,1]$). Fix $\omega\in\Omega_{0}$ and write $r=\varsigma(\omega)$.

If $r=T$: the two time integrals and the integral in $R^{M,\varsigma}$ are $0$ by convention, $M^{\gamma}_{T}-M^{\gamma}_{\varsigma}=0$, so $R^{M,\varsigma}=0$, and the identity reduces to $G(\Sigma_{T})-G(S_{T})+\sum_{\gamma}P^{\gamma}_{T}y^{\gamma}_{T}=\mathcal{D}_{G}$, which holds because $G=\bar{G}$ on $\Delta^{l}$, $P^{\gamma}_{T}=-\partial_{\gamma}\bar{G}(S_{T})$, and $\mathcal{D}_{G}=\bar{G}(\Sigma_{T})-\bar{G}(S_{T})-\sum_{\gamma}\partial_{\gamma}\bar{G}(S_{T})\,y^{\gamma}_{T}$.

Assume $r<T$. By part (b) of the martingale decomposition theorem at the times $t$ and $r$ (with $\mathbf{1}_{\Omega_{0}}(\omega)=1$) and (0.2), for every $t\in[r,T]$ and every $\gamma$,

$$\Sigma^{\gamma}_{t}-\Sigma^{\gamma}_{r}=\int_{[r,t]}b^{\gamma}(\Sigma_{s},\alpha_{s})\,ds+M^{\gamma}_{t}-M^{\gamma}_{r},\qquad S^{\gamma}_{t}-S^{\gamma}_{r}=\int_{[r,t]}b^{\gamma}(S_{s},A_{s})\,ds,$$

the second by clause 2 of the trajectory-pair definition, claim 3 of the toolkit, and Riemann additivity (or trivially when $r=0$ or $t=r$); subtracting,

$$y^{\gamma}_{t}=y^{\gamma}_{r}+\int_{[r,t]}g^{\gamma}(s)\,ds+M^{\gamma}_{t}-M^{\gamma}_{r}\qquad(t\in[r,T]).\tag{2.1}$$

The path $s\mapsto g^{\gamma}(s)$ is measurable and bounded by $4(l-1)B$ (part (a) of the decomposition theorem for the first term; continuity for the second). Define $w^{\gamma}_{t}=y^{\gamma}_{r}+\int_{[0,t]}\mathbf{1}_{\{s\ge r\}}\,g^{\gamma}(s)\,ds$ for $t\in[0,T]$; the integrand is measurable (arithmetic of measurable functions, the indicator being that of the Borel set $[r,T]$) and bounded, and by (0.3) and (0.2), $w^{\gamma}_{t}=y^{\gamma}_{r}$ for $t\le r$ while $w^{\gamma}_{t}=y^{\gamma}_{r}+\int_{[r,t]}g^{\gamma}(s)\,ds=y^{\gamma}_{t}-\big(M^{\gamma}_{t}-M^{\gamma}_{r}\big)$ for $t\in[r,T]$, by (2.1). Apply part (ii) of integration by parts for indefinite Lebesgue integrals with $u_{0}=P^{\gamma}_{0}$, $f=f^{\gamma}$ (Step 1), $v_{0}=y^{\gamma}_{r}$ and $g=\mathbf{1}_{\{\cdot\ge r\}}g^{\gamma}$:

$$P^{\gamma}_{T}\,w^{\gamma}_{T}=P^{\gamma}_{0}\,y^{\gamma}_{r}+\int_{[0,T]}\Big(f^{\gamma}(s)\,w^{\gamma}_{s}+P^{\gamma}_{s}\,\mathbf{1}_{\{s\ge r\}}\,g^{\gamma}(s)\Big)ds .$$

By (0.2) and (0.3), using $w^{\gamma}_{s}=y^{\gamma}_{r}$ on $[0,r]$ (the values at the single point $r$ agreeing) and Step 1,

$$\int_{[0,T]}f^{\gamma}w^{\gamma}\,ds=\big(P^{\gamma}_{r}-P^{\gamma}_{0}\big)y^{\gamma}_{r}+\int_{[r,T]}f^{\gamma}(s)\,y^{\gamma}_{s}\,ds-\int_{[r,T]}f^{\gamma}(s)\big(M^{\gamma}_{s}-M^{\gamma}_{r}\big)ds,\qquad \int_{[0,T]}P^{\gamma}\mathbf{1}_{\{\cdot\ge r\}}g^{\gamma}\,ds=\int_{[r,T]}P^{\gamma}_{s}\,g^{\gamma}(s)\,ds,$$

and $w^{\gamma}_{T}=y^{\gamma}_{T}-(M^{\gamma}_{T}-M^{\gamma}_{r})$. Substituting and rearranging,

$$\int_{[r,T]}P^{\gamma}_{s}\,g^{\gamma}(s)\,ds=P^{\gamma}_{T}y^{\gamma}_{T}-P^{\gamma}_{r}y^{\gamma}_{r}-\int_{[r,T]}f^{\gamma}y^{\gamma}\,ds\;-\;P^{\gamma}_{T}\big(M^{\gamma}_{T}-M^{\gamma}_{r}\big)+\int_{[r,T]}f^{\gamma}\big(M^{\gamma}_{s}-M^{\gamma}_{r}\big)ds .\tag{2.2}$$

By the two pointwise identities of the preamble and linearity,

$$\int_{[r,T]}\big(L(\Sigma_{t},\alpha_{t})-L(S_{t},A_{t})\big)dt=\int_{[r,T]}\mathcal{D}_{t}\,dt+\sum_{\gamma=1}^{l}\int_{[r,T]}f^{\gamma}y^{\gamma}\,dt+\sum_{\gamma=1}^{l}\int_{[r,T]}P^{\gamma}g^{\gamma}\,dt ,$$

and summing (2.2) over $\gamma$, the $\int f^{\gamma}y^{\gamma}$ terms cancel, giving

$$\int_{[r,T]}\big(L(\Sigma_{t},\alpha_{t})-L(S_{t},A_{t})\big)dt=\int_{[r,T]}\mathcal{D}_{t}\,dt+\sum_{\gamma}P^{\gamma}_{T}y^{\gamma}_{T}-\sum_{\gamma}P^{\gamma}_{r}y^{\gamma}_{r}+R^{M,\varsigma}.$$

Adding $G(\Sigma_{T})-G(S_{T})=\bar{G}(\Sigma_{T})-\bar{G}(S_{T})=\mathcal{D}_{G}+\sum_{\gamma}\partial_{\gamma}\bar{G}(S_{T})y^{\gamma}_{T}=\mathcal{D}_{G}-\sum_{\gamma}P^{\gamma}_{T}y^{\gamma}_{T}$ and then $\sum_{\gamma}P^{\gamma}_{r}y^{\gamma}_{r}$ to both sides yields the identity of claim 1. Finally, splitting the difference of integrals by (0.2) shows both $\int_{[r,T]}L(\Sigma_{t},\alpha_{t})\,dt$ and $\int_{[r,T]}L(S_{t},A_{t})\,dt$ exist separately, as claimed.

Measurability in $\omega$. The pre-stopping-time indicator $I=(I_{t})$ of $\varsigma$ is progressively measurable (claim 1 of the stopped-integral lemma), so, taking the time index $T$, the map $(t,\omega)\mapsto1-I_{t}(\omega)=\mathbf{1}_{\{\varsigma(\omega)\le t\}}$ is product-measurable. By (0.3), at every $\omega\in\Omega_{0}$,

$$\mathbf{1}_{\Omega_{0}}\int_{[\varsigma,T]}\mathcal{D}_{t}\,dt=\int_{[0,T]}\big(1-I_{t}\big)\,\mathbf{1}_{\Omega_{0}}\,\mathcal{D}_{t}\,dt ,$$

and the right-hand side vanishes at $\omega\notin\Omega_{0}$; the integrand is product-measurable (products of measurable maps being again measurable, as sequentially continuous functions of them) and bounded by $C_{\mathcal{D}}$, so splitting it into positive and negative parts and applying the Tonelli theorem (the trace Lebesgue measure and $P$ being finite), the right-hand side is the difference of two measurable $[0,\infty]$-valued functions of $\omega$ that are bounded by $C_{\mathcal{D}}T$, hence a bounded random variable. The same argument applies to $\int_{[0,T]}(1-I_{t})\,f^{\gamma}(t)\,\mathbf{1}_{\Omega_{0}}M^{\gamma}_{t}\,dt$. Next, $\mathbf{1}_{\Omega_{0}}M^{\gamma}$ and $\mathbf{1}_{\Omega_{0}}\Sigma^{\gamma}$ are progressively measurable, so their sampled functions at $\varsigma$ are $\mathcal{F}_{\varsigma}$-measurable random variables (claim 4(ii) of the stopping-time toolkit), and these equal $\mathbf{1}_{\Omega_{0}}M^{\gamma}_{\varsigma}$ and $\mathbf{1}_{\Omega_{0}}\Sigma^{\gamma}_{\varsigma}$ pointwise; $\varsigma$ is a random variable (claim 1 of the same toolkit), so $\omega\mapsto P^{\gamma}_{\varsigma(\omega)}$ and $\omega\mapsto S^{\gamma}_{\varsigma(\omega)}$ are random variables by measurability of sequentially continuous functions of measurable maps ($P^{\gamma}$ and $S^{\gamma}$ being continuous on $[0,T]$). Hence $\mathbf{1}_{\Omega_{0}}y^{\gamma}_{\varsigma}$ and $\mathbf{1}_{\Omega_{0}}|\mathfrak{s}_{\varsigma}|^{2}=N\sum_{\gamma}(\mathbf{1}_{\Omega_{0}}y^{\gamma}_{\varsigma})^{2}$ are bounded random variables. Finally, at every $\omega\in\Omega_{0}$ (using (0.3) at $r=\varsigma(\omega)$, Step 1, and the constancy of $M^{\gamma}_{\varsigma}(\omega)$ in $t$),

$$R^{M,\varsigma}=\sum_{\gamma=1}^{l}\Big(-P^{\gamma}_{T}\big(M^{\gamma}_{T}-M^{\gamma}_{\varsigma}\big)+\int_{[0,T]}\big(1-I_{t}\big)f^{\gamma}(t)\,M^{\gamma}_{t}\,dt-M^{\gamma}_{\varsigma}\big(P^{\gamma}_{T}-P^{\gamma}_{\varsigma}\big)\Big),\tag{2.3}$$

and multiplying through by $\mathbf{1}_{\Omega_{0}}$ (which reproduces $R^{M,\varsigma}$, this being $0$ off $\Omega_{0}$) exhibits $R^{M,\varsigma}$ as a finite sum and product of the bounded random variables just listed; $|R^{M,\varsigma}|\le2K'_{M}\,(C_{P}+C_{\partial}T)$ everywhere, since $|M^{\gamma}_{t}-M^{\gamma}_{\varsigma}|\le2K'_{M}$ while $\sum_{\gamma}|P^{\gamma}_{T}|\le C_{P}$ and $\sum_{\gamma}|f^{\gamma}(t)|\le C_{\partial}$.

Step 3: proof of claim 2. Fix $\gamma$ and let $D\in\mathcal{F}_{\varsigma}$. Since $R^{M,\varsigma}$ vanishes off $\Omega_{0}$, we may work with (2.3) multiplied by $\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}$. The process $M^{\gamma}$ satisfies the hypotheses of the optional stopping theorem for the filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$: it is a square-integrable martingale by part (b) of the decomposition theorem, $|M^{\gamma}_{t}|\le K'_{M}$ everywhere, and its paths are right-continuous at every $t\in[0,T)$ in the required sense at every point of the probability-one event $\Omega_{0}$.

(i) Terminal term. The constant $T$ is a stopping time (claim 1 of the stopping-time toolkit) and $\varsigma\le T$ pointwise, so part (b) of the optional stopping theorem with $\sigma=\varsigma$, $\tau=T$ and the event $D\in\mathcal{F}_{\varsigma}$ gives $\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}(M^{\gamma}_{T}-M^{\gamma}_{\varsigma})]=0$.

(ii) The set $D\cap\{\varsigma\le t\}$. Fix $t\in[0,T]$ and put $D''=D\cap\{\varsigma\le t\}$. We claim $D''\in\mathcal{F}_{\min(\varsigma,t)}$, the $\sigma$-algebra prior to the stopping time $\min(\varsigma,t)$ (a stopping time by claim 1 of the toolkit). Indeed, for $s\in[0,T]$: if $s\ge t$ then $D''\cap\{\min(\varsigma,t)\le s\}=D''=D\cap\{\varsigma\le t\}\in\mathcal{F}_{t}\subseteq\mathcal{F}_{s}$ (by the definition of $\mathcal{F}_{\varsigma}$ at the time $t$, and the filtration property); while if $s<t$ then $\{\min(\varsigma,t)\le s\}=\{\varsigma\le s\}$ and $D''\cap\{\varsigma\le s\}=D\cap\{\varsigma\le s\}\in\mathcal{F}_{s}$ (the definition of $\mathcal{F}_{\varsigma}$ at the time $s$, and $\{\varsigma\le s\}\subseteq\{\varsigma\le t\}$).

(iii) One time slice. Part (b) of the optional stopping theorem with $\sigma=\min(\varsigma,t)$, $\tau=t$ (constant; $\min(\varsigma,t)\le t$ pointwise) and the event $D''\in\mathcal{F}_{\sigma}$ gives $\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D''}M^{\gamma}_{t}]=\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D''}M^{\gamma}_{\min(\varsigma,t)}]$, and $M^{\gamma}_{\min(\varsigma,t)}=M^{\gamma}_{\varsigma}$ at every point of $D''$; hence

$$\mathbb{E}\big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\,\mathbf{1}_{\{\varsigma\le t\}}\,M^{\gamma}_{t}\big]=\mathbb{E}\big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\,\mathbf{1}_{\{\varsigma\le t\}}\,M^{\gamma}_{\varsigma}\big]\qquad(t\in[0,T]).\tag{3.1}$$

(iv) The integral term. The map $(t,\omega)\mapsto\mathbf{1}_{D}\mathbf{1}_{\Omega_{0}}(1-I_{t})f^{\gamma}(t)M^{\gamma}_{t}$ is product-measurable and bounded ($f^{\gamma}$ continuous, hence its extension $(t,\omega)\mapsto f^{\gamma}(t)$ product-measurable as a sequentially continuous function of $(t,\omega)\mapsto t$, whose preimages of Borel sets are rectangles), and likewise with $M^{\gamma}_{t}$ replaced by the random variable $\mathbf{1}_{\Omega_{0}}M^{\gamma}_{\varsigma}$ (constant in $t$; preimages are rectangles). By the Fubini theorem for bounded (hence integrable) product-measurable functions over the finite product measure, together with (3.1),

$$\mathbb{E}\Big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\int_{[0,T]}(1-I_{t})f^{\gamma}(t)M^{\gamma}_{t}\,dt\Big]=\int_{[0,T]}f^{\gamma}(t)\,\mathbb{E}\big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\mathbf{1}_{\{\varsigma\le t\}}M^{\gamma}_{t}\big]dt=\int_{[0,T]}f^{\gamma}(t)\,\mathbb{E}\big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\mathbf{1}_{\{\varsigma\le t\}}M^{\gamma}_{\varsigma}\big]dt=\mathbb{E}\Big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\,M^{\gamma}_{\varsigma}\int_{[0,T]}\mathbf{1}_{\{\varsigma\le t\}}f^{\gamma}(t)\,dt\Big]=\mathbb{E}\big[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}\,M^{\gamma}_{\varsigma}\big(P^{\gamma}_{T}-P^{\gamma}_{\varsigma}\big)\big],$$

the last equality by (0.3) and Step 1 at $r=\varsigma(\omega)$, pointwise in $\omega$. Taking the expectation of $\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}$ times (2.3) and inserting (i) and the display above, the three contributions cancel for each $\gamma$: $\mathbb{E}[\mathbf{1}_{D}R^{M,\varsigma}]=\sum_{\gamma}\big(0+\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}M^{\gamma}_{\varsigma}(P^{\gamma}_{T}-P^{\gamma}_{\varsigma})]-\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathbf{1}_{D}M^{\gamma}_{\varsigma}(P^{\gamma}_{T}-P^{\gamma}_{\varsigma})]\big)=0$.

Step 4: proof of claim 3. For each $\gamma$ let $\overline{M}^{\gamma}$ be the random variable furnished by the supremum lemma for the bounded process $|M^{\gamma}_{t}|\mathbf{1}_{\Omega_{0}}$ (whose paths are right-continuous at every point of $\Omega_{0}$), so that $0\le\overline{M}^{\gamma}\le K'_{M}$ and $\overline{M}^{\gamma}(\omega)=\sup_{t\in[0,T]}|M^{\gamma}_{t}(\omega)|$ for $\omega\in\Omega_{0}$, and put $\hat{M}=\sum_{\gamma=1}^{l}\overline{M}^{\gamma}$. By the fourth-moment maximal inequality applied to the martingale $M^{\gamma}$ (its hypotheses verified in Step 3) and part (b) of the restricted moments lemma, using $(M^{\gamma}_{T})^{4}\le|M_{T}|^{4}$ (claim 1 of the componentwise toolkit and monotonicity of squaring on nonnegative reals),

$$\mathbb{E}\big[(\overline{M}^{\gamma})^{4}\big]\le4\,\mathbb{E}\big[(M^{\gamma}_{T})^{4}\big]\le4\,c_{M}\,\kappa_{T}\,N^{-2}.\tag{4.1}$$

(4a) A pointwise lower bound on $D$. Let $\omega\in D$, so $\omega\in\Omega_{0}$ and $|y_{r}|\le\varepsilon_{tg}$ with $r=\varsigma(\omega)$, where $y_{r}$ abbreviates $y_{\varsigma(\omega)}(\omega)$. We show

$$\int_{[\varsigma,T]}\mathcal{D}_{t}\,dt+\mathcal{D}_{G}\ \ge\ -C^{\vee}_{tg}\,|y_{r}|^{2}-2\,(K_{L}T+K_{G})\,e^{l\Lambda_{b}T}\,\hat{M}(\omega)-R^{M,\varsigma}(\omega).\tag{4.2}$$

If $r=T$: the time integral and $R^{M,\varsigma}$ vanish (Step 2), $\hat{M}\ge0$, and $y_{r}=y_{T}$. The second display of conclusion (b) of the first-order expansion lemma for the mean-field cost, applied at $\Sigma=\Sigma_{T}(\omega)\in\Delta^{l}$ — its hypotheses being the compactness and convexity of $\mathcal{A}$, part of the data of the affine-controlled family, together with the stationary triple already fixed — bounds exactly the quantity $\mathcal{D}_{G}=\bar{G}(\Sigma_{T})-\bar{G}(S_{T})-\sum_{\gamma}\partial_{\gamma}\bar{G}(S_{T})y^{\gamma}_{T}$:

$$|\mathcal{D}_{G}|\le\tfrac{l\,K_{c}}{2}\,|y_{T}|^{2},\qquad\text{hence}\qquad \mathcal{D}_{G}\ \ge\ -\tfrac{l\,K_{c}}{2}\,|y_{r}|^{2}\ \ge\ -C^{\vee}_{tg}\,|y_{r}|^{2}$$

since $C^{\vee}_{tg}\ge\tfrac{l\,K_{c}}{2}$; so (4.2) holds. (Only this hypothesis-free bound on the terminal remainder is used; the coercive part (d) of the recentred expansion lemma, which additionally assumes (H1) and (U), is not invoked anywhere in this proof.) Assume now $r<T$ and write $T^{\sharp}=T-r$.

The shifted realized control. By claim 3 of the realized-control lemma, the path $p:t\mapsto\hat{\alpha}(t,\omega)$ on $[0,T]$ is measurable with every value in $\mathcal{A}$, and $\hat{\alpha}(t,\omega)=\alpha_{t}(\omega)$ for every $t$ (claim 2 there, $\omega\in\Omega_{0}$). By the measurability part of (0.1), the shifted path $a^{\sharp}:t\mapsto p_{r+t}$ on $[0,T^{\sharp}]$ is measurable; it takes values in $\mathcal{A}$, is bounded (by the constant $R$ of the realized-control lemma), hence its components are square-integrable, so $a^{\sharp}\in\mathcal{L}^{2}([0,T^{\sharp}];\mathbb{R}^{m})$ and, by the definition of the control set, its class $\eta=[a^{\sharp}]$ lies in $\mathcal{U}^{[T^{\sharp}]}_{\mathcal{A}}$ with admissible representative $a^{\sharp}$.

The per-path mean-field flow and its tracking. Let $x^{\natural}=S^{[T^{\sharp}]}(\Sigma_{r}(\omega),\eta)$, by claim 2 of the flow stability lemma (horizon-$T^{\sharp}$ instance, admissible representative $a^{\sharp}$; $\Sigma_{r}(\omega)\in\Delta^{l}$ by the solution definition) the continuous $\Delta^{l}$-valued solution of $x^{\natural\gamma}_{t}=\Sigma^{\gamma}_{r}+\int_{[0,t]}\hat{b}^{\gamma}(x^{\natural}_{s},a^{\sharp}_{s})\,ds$, with $\hat{b}^{\gamma}(x^{\natural}_{s},a^{\sharp}_{s})=b^{\gamma}(x^{\natural}_{s},a^{\sharp}_{s})$ by claim 6 of the affine rate-family lemma. Set $e_{t}=\Sigma_{r+t}(\omega)-x^{\natural}_{t}$ for $t\in[0,T^{\sharp}]$. By the first display of Step 2 at the time $r+t$, shifted with (0.1) and using $\alpha_{s}(\omega)=p_{s}$ for every $s$ and $p_{r+s}=a^{\sharp}_{s}$ — so that $\Sigma^{\gamma}_{r+t}=\Sigma^{\gamma}_{r}+\int_{[0,t]}b^{\gamma}(\Sigma_{r+s},a^{\sharp}_{s})\,ds+M^{\gamma}_{r+t}-M^{\gamma}_{r}$ — subtracting the flow equation and using the scalar integral triangle inequality componentwise,

$$|e^{\gamma}_{t}|\le\big|M^{\gamma}_{r+t}-M^{\gamma}_{r}\big|+\int_{[0,t]}\big|b^{\gamma}(\Sigma_{r+s},a^{\sharp}_{s})-b^{\gamma}(x^{\natural}_{s},a^{\sharp}_{s})\big|\,ds .$$

Summing over $\gamma$ and using claim 1 of the componentwise toolkit twice ($|v^{\gamma}|\le|v|$ and $\sum_{\gamma}|v^{\gamma}|\le l\max_{\gamma}|v^{\gamma}|\le l\,|v|$) together with the state-Lipschitz bound $|b(\Sigma,a)-b(\Sigma',a)|\le\Lambda_{b}|\Sigma-\Sigma'|$ of claim 4 of the affine rate-family lemma, the function $u(t)=\sum_{\gamma}|e^{\gamma}_{t}|$ — measurable in $t$ (paths of $\Sigma$ measurable, $x^{\natural}$ continuous, absolute values and sums of measurable functions measurable) and bounded by $2l$ — satisfies

$$u(t)\le2\,\hat{M}(\omega)+l\,\Lambda_{b}\int_{[0,t]}u(s)\,ds\qquad(t\in[0,T^{\sharp}]),$$

since $\sum_{\gamma}|M^{\gamma}_{r+t}-M^{\gamma}_{r}|\le2\hat{M}(\omega)$ and $|e_{s}|\le u(s)$. By Gronwall's lemma for bounded measurable functions (horizon $T^{\sharp}$), $u(t)\le2\hat{M}(\omega)\,e^{l\Lambda_{b}t}$, so

$$|e_{t}|\le u(t)\le2\,\hat{M}(\omega)\,e^{l\Lambda_{b}T}\qquad(t\in[0,T^{\sharp}]).\tag{4.3}$$

Comparison of costs. By (0.1) and (0.2), $\int_{[r,T]}L(\Sigma_{t},\alpha_{t})\,dt=\int_{[0,T^{\sharp}]}L(\Sigma_{r+t},a^{\sharp}_{t})\,dt$. By the definition of the mean-field cost for the horizon-$T^{\sharp}$ instance with the admissible representative $a^{\sharp}$ (its provisions recording that the integrand below is measurable and bounded), $F^{[T^{\sharp}]}(\Sigma_{r},\eta)=\int_{[0,T^{\sharp}]}L(x^{\natural}_{t},a^{\sharp}_{t})\,dt+G(x^{\natural}_{T^{\sharp}})$. By (LipC), (4.3) and the scalar integral triangle inequality,

$$\Big|\int_{[0,T^{\sharp}]}L(\Sigma_{r+t},a^{\sharp}_{t})\,dt+G(\Sigma_{T})-F^{[T^{\sharp}]}(\Sigma_{r},\eta)\Big|\le K_{L}\int_{[0,T^{\sharp}]}|e_{t}|\,dt+K_{G}|e_{T^{\sharp}}|\le2\,(K_{L}T+K_{G})\,e^{l\Lambda_{b}T}\,\hat{M}(\omega),$$

using $\Sigma_{T}(\omega)=\Sigma_{r+T^{\sharp}}(\omega)$ and $\lambda_{[0,T^{\sharp}]}([0,T^{\sharp}])=T^{\sharp}\le T$. By conclusion 2 of the to-go comparison lemma — applicable at $t_{0}=r\in[0,T)$, $x=\Sigma_{r}(\omega)\in\Delta^{l}$ with $|x-S_{r}|\le\varepsilon_{tg}$, and $\xi=\eta$, under the assumed (LipC)-independent hypotheses $[A]\in\mathcal{M}^{*}_{S_{0}}$ and (TG) —

$$F^{[T^{\sharp}]}(\Sigma_{r},\eta)\ \ge\ \int_{[r,T]}L(S_{t},A_{t})\,dt+G(S_{T})-\sum_{\gamma=1}^{l}P^{\gamma}_{r}\,y^{\gamma}_{r}-C_{tg}\,|y_{r}|^{2}.$$

Combining the last two displays with the identity of claim 1 (and $\alpha_{t}(\omega)=p_{t}$ for every $t$),

$$\int_{[\varsigma,T]}\mathcal{D}_{t}\,dt+\mathcal{D}_{G}\ \ge\ -C_{tg}|y_{r}|^{2}-2(K_{L}T+K_{G})e^{l\Lambda_{b}T}\hat{M}(\omega)-R^{M,\varsigma}(\omega),$$

which implies (4.2) because $C_{tg}\le C^{\vee}_{tg}$.

(4b) Expectations. By (4.2), the function

$$Z=\mathbf{1}_{D}\Big(\int_{[\varsigma,T]}\mathcal{D}_{t}\,dt+\mathcal{D}_{G}\Big)+C^{\vee}_{tg}\,\mathbf{1}_{D}\,|y_{\varsigma}|^{2}+2(K_{L}T+K_{G})e^{l\Lambda_{b}T}\,\mathbf{1}_{D}\,\hat{M}+\mathbf{1}_{D}\,R^{M,\varsigma}$$

is nonnegative at every point of $\Omega$ (every term vanishes off $D$, and on $D$ the nonnegativity is exactly (4.2)), and it is a bounded random variable: the first and last terms by claim 1 ($\mathcal{D}_{G}$ is a random variable bounded by $C_{\mathcal{D}G}$ at every point, by part (a) of the expansion lemma and the solution definition, and $D\subseteq\Omega_{0}$ allows the insertion of $\mathbf{1}_{\Omega_{0}}$ throughout), the middle terms by claim 1 and Step 4's construction of $\hat{M}$. Hence $\mathbb{E}[N\,Z]\ge0$, and by linearity, claim 2 (with the event $D\in\mathcal{F}_{\varsigma}$), and $N|y_{\varsigma}|^{2}=|\mathfrak{s}_{\varsigma}|^{2}$ on $\Omega_{0}$,

$$\mathbb{E}\Big[\mathbf{1}_{D}\Big(N\int_{[\varsigma,T]}\mathcal{D}_{t}\,dt+N\mathcal{D}_{G}\Big)\Big]\ \ge\ -C^{\vee}_{tg}\,\mathbb{E}\big[\mathbf{1}_{D}|\mathfrak{s}_{\varsigma}|^{2}\big]-2(K_{L}T+K_{G})e^{l\Lambda_{b}T}\,N\,\mathbb{E}\big[\mathbf{1}_{D}\hat{M}\big].$$

By part (a) of the restricted moments lemma applied to each $\overline{M}^{\gamma}$, (4.1), and the multiplicativity and monotonicity of $x\mapsto x^{1/4}$ recorded there,

$$\mathbb{E}\big[\mathbf{1}_{D}\hat{M}\big]=\sum_{\gamma=1}^{l}\mathbb{E}\big[\mathbf{1}_{D}\overline{M}^{\gamma}\big]\le\sum_{\gamma=1}^{l}P(D)^{3/4}\,\mathbb{E}\big[(\overline{M}^{\gamma})^{4}\big]^{1/4}\le l\,\big(4c_{M}\kappa_{T}\big)^{1/4}\,N^{-1/2}\,P(D)^{3/4},$$

and $N\cdot N^{-1/2}=\sqrt{N}$, which gives the bound of claim 3 with $C_{ns}=2\,l\,(K_{L}T+K_{G})\,e^{l\Lambda_{b}T}(4c_{M}\kappa_{T})^{1/4}$. $\blacksquare$