Observation-Measurability of the Cascade Good Sets and the Restricted Filtering Bound

Adopt the setting and notation of the block cascade lemma — the solution of the controlled $N$-agent dynamics with regular event $\Omega_{0}$, empirical state measure $\Sigma$, control $\alpha$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$; the block data $T_{0}$, $K$, $t_{k}$, the levels $L_{k}$, the anchored clocks $\sigma^{(k)}$, the good sets $G_{k}$ and the leave events $D_{k}$ — and the setting and notation of the completion-of-squares theorem: the fluctuation processes $\mathfrak{s}_{t}$, $\mathfrak{a}_{t}$ about a mean-field trajectory pair $(S,A)$, the matrices $R_{t}$ and $W_{t}$, hypothesis (H1) with constant $r>0$, hypothesis (H2) with Riccati family $Z$, and the process $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{T}\mathfrak{s}_{t}$. Write $\mathbb{E}$ for the expectation, $\mathbf{1}_{D}$ for the indicator of a set $D$, $x\cdot My=\sum_{p,q}M^{pq}x^{p}y^{q}$ for the entry pairing of the completion-of-squares theorem, and $|\cdot|$ for the Euclidean norm. Set

$$\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{T},$$

a real matrix with $l$ rows and $l$ columns, as in conclusion (b) of the filtering lower-bound reduction lemma.

Then the following hold.

0. (The regular event is observable.) $\Omega_{0}\in\mathcal{G}_{0}$.

1. (The good sets are observable.) For every $k\in\{0,\dots,K\}$ the event $G_{k}$ belongs to $\mathcal{G}_{t_{k}}$, and for every $k\in\{0,\dots,K-1\}$ and every $t\in[0,T]$ the tracked event

$$\mathcal{T}_{k}(t)=G_{k}\cap\{t<\sigma^{(k)}\}$$

belongs to $\mathcal{G}_{t}$ whenever $t\ge t_{k}$.

2. (Symmetry and positive semidefiniteness of $\Xi$.) For every $t\in[0,T]$ the matrix $\Xi_{t}$ is symmetric and positive semidefinite, and its entries are continuous, hence bounded, on $[0,T]$.

3. (Restricted filtering bound.) Let $t\in[0,T]$ and let $\mathcal{H}\in\mathcal{G}_{t}$ be an event. The components of $\mathfrak{s}_{t}$ and of $\mathfrak{a}_{t}$ are square-integrable, each component of $\mathfrak{a}_{t}$ is almost surely equal to a $\mathcal{G}_{t}$-measurable square-integrable random variable by the observation-adaptedness lemma, and for any choice of conditional expectations $\mathbb{E}[\mathfrak{s}^{\gamma}_{t}\mid\mathcal{G}_{t}]$, which exist by the existence and uniqueness theorem, the filtering error $\varepsilon_{t}$ with components $\varepsilon^{\gamma}_{t}=\mathfrak{s}^{\gamma}_{t}-\mathbb{E}[\mathfrak{s}^{\gamma}_{t}\mid\mathcal{G}_{t}]$ satisfies

$$\mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}\,u_{t}\cdot R_{t}u_{t}\bigr]\ \ge\ \sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\Xi^{\gamma\delta}_{t}\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{H}}\,\varepsilon^{\gamma}_{t}\varepsilon^{\delta}_{t}\bigr]\ \ge\ 0,$$

all the expectations being finite. The middle quantity is unchanged if the conditional expectations $\mathbb{E}[\mathfrak{s}^{\gamma}_{t}\mid\mathcal{G}_{t}]$ are replaced by any other conditional expectations of the $\mathfrak{s}^{\gamma}_{t}$ given $\mathcal{G}_{t}$.

4. (Application to the cascade.) In particular, for every $k\in\{0,\dots,K-1\}$ and every $t\in[t_{k},T]$,

$$\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\mathbf{1}_{\{t<\sigma^{(k)}\}}\,u_{t}\cdot R_{t}u_{t}\bigr]\ \ge\ \sum_{\gamma,\delta=1}^{l}\Xi^{\gamma\delta}_{t}\,\mathbb{E}\bigl[\mathbf{1}_{G_{k}}\mathbf{1}_{\{t<\sigma^{(k)}\}}\,\varepsilon^{\gamma}_{t}\varepsilon^{\delta}_{t}\bigr]\ \ge\ 0 .$$