Conditional-Variance Lower Bound for the Completed-Square Control Energy

Adopt the setting, hypotheses (H1)--(H2), and notation of the completion-of-squares theorem for the fluctuation cost: in particular the fluctuation processes $\mathfrak{s}_t$ and $\mathfrak{a}_t$ of a solution of the controlled $N$-agent dynamics about a mean-field trajectory pair $(S,A)$, the symmetric positive definite matrices $R_t$ (conclusion (a) of that theorem, under (H1)) with inverses $R_t^{-1}$ (existing, symmetric, and positive definite), the Riccati family $Z$ of (H2) with $W_t=Z_t\mathsf{B}_t+\tfrac{1}{2}V_t$, the completed-square control deviation $u_t=\mathfrak{a}_t+R_t^{-1}W_t^{\top}\mathfrak{s}_t$, and the matrix entry notation $x\cdot My=\sum_{p,q}M^{pq}x^py^q$, $(MM')^{pq}=\sum_{r}M^{pr}M'^{rq}$, $(M^{\top})^{qp}=M^{pq}$. Let $(\mathcal{G}_t)_{t\in[0,T]}$ be the observation filtration of the solution definition, as in the adaptedness lemma, and write $\mathbb{E}$ for the expectation.

Fix $t\in[0,T]$ and assume that each component $\mathfrak{a}^j_t$ ($j\in\{1,\dots,m\}$) is square-integrable. Each component $\mathfrak{s}^\gamma_t$ is square-integrable, being a random variable with $(\mathfrak{s}^\gamma_t)^2\le2N$ everywhere (the probability simplex having squared Euclidean diameter at most $2$). By the existence and uniqueness theorem for conditional expectation, for each $\gamma\in\{1,\dots,l\}$ there is a conditional expectation $\hat{\mathfrak{s}}^\gamma_t$ of $\mathfrak{s}^\gamma_t$ given $\mathcal{G}_t$; fix such a choice, write $\hat{\mathfrak{s}}_t=(\hat{\mathfrak{s}}^1_t,\dots,\hat{\mathfrak{s}}^l_t)$, and set $\varepsilon_t=\mathfrak{s}_t-\hat{\mathfrak{s}}_t$ componentwise. Then:

1. (Conditional-variance lower bound.) The random variables $u_t\cdot R_tu_t$ and $\varepsilon_t\cdot(W_tR_t^{-1}W_t^{\top})\varepsilon_t$ are integrable, and

$$\mathbb{E}\big[u_t\cdot R_tu_t\big]\ \ge\ \mathbb{E}\big[\varepsilon_t\cdot(W_tR_t^{-1}W_t^{\top})\,\varepsilon_t\big].$$

2. (Trace form and independence of the choice.)

$$\mathbb{E}\big[\varepsilon_t\cdot(W_tR_t^{-1}W_t^{\top})\varepsilon_t\big]=\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}\big(W_tR_t^{-1}W_t^{\top}\big)^{\gamma\delta}\,\mathbb{E}\big[\varepsilon^\gamma_t\,\varepsilon^\delta_t\big],$$

and each $\mathbb{E}[\varepsilon^\gamma_t\varepsilon^\delta_t]$ is unchanged if the conditional expectations $\hat{\mathfrak{s}}^\gamma_t$ are replaced by any other conditional expectations of the $\mathfrak{s}^\gamma_t$ given $\mathcal{G}_t$.