Stopped Completion of Squares on a Cascade Block

Adopt the setting, notation and hypotheses of the completion-of-squares theorem for the fluctuation cost: the fluctuation processes $\mathfrak{s}_{t}$, $\mathfrak{a}_{t}$ of a solution with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$, taken about a mean-field trajectory pair $(S,A)$; the extension $(U,V,\bar{\beta})$ with derivative bound $K$ and extended aggregate state drift $\bar{b}$; the extension $(U_{c},\bar{L},\bar{G})$ with second-derivative bound $K_{c}$; the stationary co-state $P$ with bound $C_{P}$; the fluctuation Hessian coefficients $H_{ij}(t)$ and $F_{\gamma\delta}$; the aggregate fluctuation covariance $\Theta$ and the aggregate state drift $b$; the drift difference $g_{s}=\sqrt{N}(b(\Sigma_{s},\alpha_{s})-b(S_{s},A_{s}))$; the matrices $E_{t},\mathsf{B}_{t},Q_{t},V_{t},R_{t},\hat{F}$ and the pairing notation $x\cdot My$ defined there; hypotheses (H1) with constant $r>0$ and (H2) with the Riccati family $Z=(Z_{t})_{t\in[0,T]}$, $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac{1}{2}V_{t}$ and densities $\dot{z}(t)=-\bigl(E_{t}^{T}Z_{t}+Z_{t}E_{t}-W_{t}R_{t}^{-1}W_{t}^{T}+Q_{t}\bigr)$; the constant $C_{Z}$ with $|Z^{\gamma\delta}_{t}|\le C_{Z}$ for all indices and all $t$, and the constant $c_{e}$; and the processes

$$u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{T}\mathfrak{s}_{t},\qquad e_{s}=g_{s}-E_{s}\mathfrak{s}_{s}-\mathsf{B}_{s}\mathfrak{a}_{s}.$$

The hypothesis $\mathcal{A}_{2}=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}_{t}|^{2}]\,dt<\infty$ of the second-order expansion theorem is part of the setting adopted here and is not re-derived; in the intended application the control set is compact, so that $|\mathfrak{a}_{t}|\le2\sqrt{N}R_{\mathcal{A}}$ everywhere with $R_{\mathcal{A}}=\sup_{a\in\mathcal{A}}|a|$ and hence $\mathcal{A}_{2}\le4R_{\mathcal{A}}^{2}TN<\infty$. The scalar $R_{\mathcal{A}}$ is distinct from the matrices $R_{t}$ above. Adopt also the weighted second-moment evolution lemma and its stopped form the stopped weighted second-moment lemma, whose weight family is the present $Z$, and, from the first-order expansion lemma for the recentred $N$-agent cost, the quantity $\mathcal{D}_{t}$ with its bound $C_{\mathcal{D}}$. Write $\mathfrak{z}_{t}=(\mathfrak{s}_{t},\mathfrak{a}_{t})\in\mathbb{R}^{l+m}$, $\mathbf{1}_{D}$ for the indicator of a set $D$, $\mathbb{E}$ for the expectation, and $\int_{[a,b]}\cdot\,ds$ for the Lebesgue integral over a compact interval. Stopping times are those of $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ and $\mathbf{1}_{\{s<\theta\}}$ denotes the pre-stopping-time indicator of a stopping time $\theta$.

Block data. Let $t_{\flat}$ and $t_{\sharp}$ be real numbers with $0\le t_{\flat}\le t_{\sharp}\le T$, let $\sigma$ be a stopping time with $\sigma(\omega)\ge t_{\flat}$ for every $\omega$, and let $\mathcal{G}$ be an event with $\mathcal{G}\subseteq\Omega_{0}$ and $\mathcal{G}\in\mathcal{F}^{\mathrm{sys}}_{t_{\flat}}$. Define $\tau:\Omega\to[0,T]$ by

$$\tau(\omega)=\sigma(\omega)\ \text{ for }\omega\in\mathcal{G},\qquad \tau(\omega)=t_{\flat}\ \text{ for }\omega\notin\mathcal{G}.$$

Then the following hold.

1. (The masked stopping time.) $\tau$ is a stopping time of $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ with $\tau\ge t_{\flat}$ everywhere; for every $s\in[t_{\flat},T]$ one has $\mathbf{1}_{\{s<\tau\}}=\mathbf{1}_{\mathcal{G}}\,\mathbf{1}_{\{s<\sigma\}}$; $\min(t_{\flat},\tau)=t_{\flat}$ everywhere; and $\min(t,\tau)=t_{\flat}$ off $\mathcal{G}$ for every $t\in[t_{\flat},T]$.

2. (Pointwise completion of squares.) At every point of $[0,T]\times\Omega$,

$$\mathfrak{s}_{s}\cdot\dot{z}(s)\,\mathfrak{s}_{s}+2\,\mathfrak{s}_{s}\cdot Z_{s}g_{s}\ =\ u_{s}\cdot R_{s}u_{s}\ -\ \tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}H_{ij}(s)\,\mathfrak{z}^{i}_{s}\mathfrak{z}^{j}_{s}\ +\ 2\,\mathfrak{s}_{s}\cdot Z_{s}e_{s}.$$

3. (Block identity.) With $\tau$ as above and $\mathfrak{s}^{\sharp}=\mathfrak{s}_{\min(t_{\sharp},\sigma)}$ denoting the sampled function taken componentwise, and $Z^{\sharp}=Z_{\min(t_{\sharp},\sigma)}$ likewise,

$$\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\Bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\;\tfrac{1}{2}\sum_{i,j=1}^{l+m}H_{ij}(s)\,\mathfrak{z}^{i}_{s}\mathfrak{z}^{j}_{s}\Bigr]ds\ =\ \mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\,\mathfrak{s}_{t_{\flat}}\cdot Z_{t_{\flat}}\mathfrak{s}_{t_{\flat}}\bigr]-\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\,\mathfrak{s}^{\sharp}\cdot Z^{\sharp}\mathfrak{s}^{\sharp}\bigr]\ +\ \int_{[t_{\flat},t_{\sharp}]}\Bigl(\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\bigl(u_{s}\cdot R_{s}u_{s}+2\,\mathfrak{s}_{s}\cdot Z_{s}e_{s}\bigr)\bigr]+\sum_{\gamma,\delta=1}^{l}Z^{\gamma\delta}_{s}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\,\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})\bigr]\Bigr)ds ,$$

all expectations and integrals being finite.

4. (Block lower bound.) Assume in addition hypothesis (JC) of the localized joint coercivity lemma, with radius $\rho^{*}$ as in its claim 2, and assume the confinement condition: $\rho_{s}(\omega)\le\rho^{*}$ whenever $s\in[t_{\flat},t_{\sharp}]$, $\omega\in\mathcal{G}$ and $s<\sigma(\omega)$. Then

$$\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\,N\mathcal{D}_{s}\bigr]ds\ \ge\ \mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\,\mathfrak{s}_{t_{\flat}}\cdot Z_{t_{\flat}}\mathfrak{s}_{t_{\flat}}\bigr]-\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\,\mathfrak{s}^{\sharp}\cdot Z^{\sharp}\mathfrak{s}^{\sharp}\bigr]+r\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}|u_{s}|^{2}\bigr]ds+\int_{[t_{\flat},t_{\sharp}]}\sum_{\gamma,\delta=1}^{l}Z^{\gamma\delta}_{s}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})\bigr]ds\ -\ \mathcal{X}^{\mathrm{lin}}\ -\ \mathcal{X}^{\mathrm{quad}},$$

where the two error terms are

$$\mathcal{X}^{\mathrm{lin}}=2\,l^{2}\,C_{Z}\,c_{e}\,N^{-1/2}\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\Bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\,|\mathfrak{s}_{s}|\bigl(|\mathfrak{s}_{s}|^{2}+|\mathfrak{a}_{s}|^{2}\bigr)\Bigr]ds,$$

and

$$\mathcal{X}^{\mathrm{quad}}=\tfrac{1}{2}(l+m)\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\Bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}\bigl(\omega_{L}(\rho_{s})+C_{P}\,\omega_{b}(\rho_{s})\bigr)\,|\mathfrak{z}_{s}|^{2}\Bigr]ds ,$$

with $\rho_{s}$ and the moduli $\omega_{L},\omega_{b}$ of the second-order expansion theorem; and moreover $\mathcal{X}^{\mathrm{quad}}\le\tfrac{c_{J}}{2}\int_{[t_{\flat},t_{\sharp}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{G}}\mathbf{1}_{\{s<\sigma\}}|\mathfrak{z}_{s}|^{2}\bigr]ds$.