Proof of Localized Joint Coercivity of the Recentred N-Agent Cost Integrand under a Positive-Definite Fluctuation Hessian

Throughout fix $t\in[0,T]$ and $\omega\in\Omega$, and write $z_{0}=(S_{t},A_{t})$ and $z=(\Sigma_{t},\alpha_{t})$, points of $\mathbb{R}^{l+m}$, with difference $w=z-z_{0}$, whose first $l$ components are $y^{\gamma}_{t}=\Sigma^{\gamma}_{t}-S^{\gamma}_{t}$ and whose last $m$ components are $\alpha^{j}_{t}-A^{j}_{t}$. By the solution definition $\Sigma_{t}\in\Delta^{l}$ and $\alpha_{t}\in\mathcal{A}$ at every point of $[0,T]\times\Omega$, and $(S_{t},A_{t})\in\Delta^{l}\times\mathcal{A}$ by the trajectory-pair definition. Note $\sqrt{N}\,w=\mathfrak{z}_{t}=(\mathfrak{s}_{t},\mathfrak{a}_{t})$ by the definition of the fluctuation processes, so that $N|w|^{2}=|\mathfrak{z}_{t}|^{2}=|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}$ and $|w|=\rho_{t}$.

Step 0: standing facts. (F1) The segment $\{z_{0}+\tau w:\tau\in[0,1]\}$ is contained in $\Delta^{l}\times\mathcal{A}$: for $\tau\in[0,1]$ the point $\tau\Sigma_{t}+(1-\tau)S_{t}$ has nonnegative coordinates summing to $1$, and $\tau\alpha_{t}+(1-\tau)A_{t}\in\mathcal{A}$ because $\mathcal{A}$ is convex by hypothesis (A). By the cost and rate extension definitions $\Delta^{l}\subset U_{c}$, $\Delta^{l}\subset U$ and $\mathcal{A}\subseteq V$, so the segment lies in the open sets $U_{c}\times\mathbb{R}^{m}$ and $U\times V$, on which $\bar{L}$ and the $\bar{b}^{\delta}$ are of class $C^{2}$ — for $\bar{b}^{\delta}$ by part (i) of the regularity of the extended aggregate state drift, for $\bar{L}$ by clause 2 of the cost extension definition. Every point of the segment lies within Euclidean distance $|w|=\rho_{t}$ of $z_{0}$, since $|(z_{0}+\tau w)-z_{0}|=\tau|w|\le|w|$. (F2) The segment $\{S_{T}+\tau(\Sigma_{T}-S_{T}):\tau\in[0,1]\}$ lies in $\Delta^{l}$, by the same coordinate computation as in (F1): for $\tau\in[0,1]$ the point $\tau\Sigma_{T}+(1-\tau)S_{T}$ has nonnegative coordinates summing to $1$, both $\Sigma_{T}$ and $S_{T}$ lying in $\Delta^{l}$. Since $\Delta^{l}\subset U_{c}$, on which $\bar{G}$ is of class $C^{2}$ by clause 2 of the cost extension definition, and since $|(S_{T}+\tau(\Sigma_{T}-S_{T}))-S_{T}|=\tau\,|\Sigma_{T}-S_{T}|\le d(\Sigma_{T},S_{T})$, every point of that segment lies in $\Delta^{l}$ within distance $d(\Sigma_{T},S_{T})$ of $S_{T}$. (F3) By clause 3 of the stationary-triple definition (stationarity), $\partial_{l+j}\bar{L}(S_{t},A_{t})-\sum_{\delta=1}^{l}P^{\delta}_{t}\,\partial_{l+j}\bar{b}^{\delta}(S_{t},A_{t})=0$ for every $j\in\{1,\dots,m\}$; that is, the partial derivatives of $\mathcal{H}_{t}$ in the last $m$ coordinates vanish at $z_{0}$. (F4) $\sum_{\delta=1}^{l}|P^{\delta}_{t}|\le C_{P}$.

Step 1: proof of claim 1. Regard $\mathcal{H}_{t}$ as the function $x\mapsto\bar{L}(x)-\sum_{\delta=1}^{l}P^{\delta}_{t}\bar{b}^{\delta}(x)$ on the open set $(U_{c}\times\mathbb{R}^{m})\cap(U\times V)$, which contains the segment of (F1); it is of class $C^{2}$ there as a finite linear combination of $C^{2}$ functions, and its partial derivatives of orders one and two at any point are the corresponding combinations, by the $C^{k}$ definition. In particular its second-order partial derivatives at $z_{0}$ are exactly the fluctuation Hessian coefficients $H_{ij}(t)=\partial_{j}\partial_{i}\bar{L}(S_{t},A_{t})-\sum_{\delta}P^{\delta}_{t}\partial_{j}\partial_{i}\bar{b}^{\delta}(S_{t},A_{t})$.

Apply part (iii) of the multivariate Taylor lemma with $n=l+m$, $x=z_{0}$, $y=z$ along the segment of (F1), separately to $f=\bar{L}$ and to each $f=\bar{b}^{\delta}$. For $\bar{L}$ the constant $\bar{\varepsilon}$ of that part may be taken to be $\omega_{L}(\rho_{t})$: every point $\tilde{z}$ of the segment lies in $\Delta^{l}\times\mathcal{A}\subseteq\Delta^{l}\times\mathbb{R}^{m}$ with $d(\tilde{z},z_{0})\le\rho_{t}$, so $|\partial_{j}\partial_{i}\bar{L}(\tilde{z})-\partial_{j}\partial_{i}\bar{L}(z_{0})|\le\omega_{L}(\rho_{t})$ by the definition of $\omega_{L}$. Similarly for $\bar{b}^{\delta}$ with $\bar{\varepsilon}=\omega_{b}(\rho_{t})$, the segment lying in $\Delta^{l}\times V$. Thus

and the same with $\bar{L}$ replaced by $\bar{b}^{\delta}$ and $\omega_{L}$ by $\omega_{b}$. Multiplying the $\bar{b}^{\delta}$ estimates by $-P^{\delta}_{t}$, summing over $\delta$, adding the $\bar{L}$ estimate, and using (F4) and the triangle inequality gives

By (F3) the first-order sum reduces to its first $l$ terms, $\sum_{\gamma=1}^{l}\partial_{\gamma}\mathcal{H}_{t}(S_{t},A_{t})\,y^{\gamma}_{t}$, so the expression inside the absolute value is exactly $\mathcal{D}_{t}-\tfrac{1}{2}\sum_{i,j}H_{ij}(t)w^{i}w^{j}$. Multiplying through by $N>0$ and using $\sqrt{N}w=\mathfrak{z}_{t}$, so that $Nw^{i}w^{j}=\mathfrak{z}^{i}_{t}\mathfrak{z}^{j}_{t}$ and $N|w|^{2}=|\mathfrak{z}_{t}|^{2}$, yields the first display of claim 1.

For the second, apply part (iii) of the Taylor lemma with $n=l$ to $f=\bar{G}$ on $U_{c}$ along the segment of (F2), with $\bar{\varepsilon}=\omega_{G}(d(\Sigma_{T},S_{T}))$, legitimate by the definition of $\omega_{G}$ since the segment lies in $\Delta^{l}$ within distance $d(\Sigma_{T},S_{T})$ of $S_{T}$; the second-order coefficients at $S_{T}$ are the $F_{\gamma\delta}$. This gives

and multiplying by $N$, with $\sqrt{N}y_{T}=\mathfrak{s}_{T}$, gives the second display. Neither argument used (JC).

Step 2: proof of claim 2. Apply part (a) of the second-order expansion theorem with the real number

(positive since $c_{J}>0$, $l+m\ge1$ and $C_{P}\ge0$): it furnishes a real $\eta>0$ with $\omega_{L}(u)\le\varepsilon$ and $\omega_{b}(u)\le\varepsilon$ for every $u\in[0,\eta]$. Put $\rho^{*}=\eta>0$. Then for $u\in[0,\rho^{*}]$,

as required.

Step 3: proof of claim 3. Let $t$ and $\omega$ satisfy $\rho_{t}\le\rho^{*}$. By claim 1 and hypothesis (JC) applied to the vector $w'=\mathfrak{z}_{t}\in\mathbb{R}^{l+m}$,

the middle step using (JC) for the first term and claim 2 at $u=\rho_{t}\in[0,\rho^{*}]$, together with $|\mathfrak{z}_{t}|^{2}\ge0$, for the second. Since $|\mathfrak{z}_{t}|^{2}=|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}$, this is the asserted inequality, and dropping the nonnegative term $\tfrac{c_{J}}{2}|\mathfrak{s}_{t}|^{2}$ gives the final sentence of claim 3.

Step 4: proof of claim 4. By claim 1 and the positive semidefiniteness of $F$ in (JC), applied to the vector $\mathfrak{s}_{T}\in\mathbb{R}^{l}$,

which is the first display of claim 4. For the second, let $\epsilon>0$ be real and apply part (a) of the second-order expansion theorem with the positive real number $2\epsilon/l$ to obtain $\rho^{*}_{G}>0$ with $\omega_{G}(u)\le2\epsilon/l$ for all $u\in[0,\rho^{*}_{G}]$; then at every $\omega$ with $d(\Sigma_{T},S_{T})\le\rho^{*}_{G}$ the first display gives $N\mathcal{D}_{G}\ge-\tfrac{1}{2}l\cdot(2\epsilon/l)\,|\mathfrak{s}_{T}|^{2}=-\epsilon|\mathfrak{s}_{T}|^{2}$.

Step 5: proof of claim 5. Let $t\in[0,T]$ and $h\in\mathbb{R}^{m}$, and apply (JC) to the vector $w=(0,h)\in\mathbb{R}^{l+m}$ whose first $l$ components vanish and whose last $m$ components are those of $h$, so that $|w|=|h|$. Since $w^{i}=0$ for $i\le l$, only the indices $i,j>l$ contribute, and writing $i=l+i'$, $j=l+j'$ with $i',j'\in\{1,\dots,m\}$,

the last equality being conclusion (b) of the quadratic growth lemma, which holds without any of the hypotheses (A), (H1), (U) and whose matrix $R_{t}$ is formed from the same fluctuation Hessian coefficients $H_{ij}$ as here. Hypothesis (JC) therefore gives $h\cdot R_{t}h\ge c_{J}|h|^{2}$, which is hypothesis (H1) with $r=c_{J}$. The final sentence of claim 5 is then conclusion (d) of that lemma, applicable under (A) — assumed in the setting adopted here — together with (H1) and (U). $\blacksquare$