Fluctuation LQG Data of a Stationary Mean-Field Triple

Let $l$, $m$, $\mathcal{A}$, $\beta$ with rate bound $B$, $(U,V,\bar{\beta})$ with derivative bound $K$, $(L,G)$, $(W,\bar{L},\bar{G})$ with second-derivative bound $K_c$, $T>0$, $(S,A)$, and the stationary co-state $P$ be as in the definition of the fluctuation linear-quadratic cost functional, with $\bar{b}$ the extended aggregate state drift of $(U,V,\bar{\beta})$, and let $H_{ij}(t)$ and $F_{\gamma\delta}$ be the fluctuation Hessian coefficients of these data. Let moreover $\tilde{l}\ge1$ be a natural number, let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, let $(\tilde{U},\bar{\tilde{\beta}})$ be a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}$, let $\bar{\tilde{b}}$ be its extended aggregate observation drift, let $\tilde{b}$ be the aggregate observation drift of $\tilde{\beta}$, and let $\Theta$ be the aggregate fluctuation covariance of $\beta$. Adopt the coordinate and partial-derivative notation of the transition-rate extension definition, so that $\partial_\gamma$ with $\gamma\le l$ differentiates in the $\gamma$-th state coordinate and $\partial_{l+j}$ with $j\le m$ in the $j$-th control coordinate, together with the notation $\partial_\gamma$ ($\gamma\in\{1,\dots,l\}$) of the observation-rate extension definition; the partial derivatives of $\bar{b}$ and of $\bar{\tilde{b}}$ used below exist by the regularity of the extended aggregate state drift and the regularity of the extended aggregate observation drift. Matrices are indexed as in the definition of the matrix-vector product: the first index labels rows and the second labels columns. We write $\mathbf{1}_{\{\cdot\}}$ for the indicator equal to $1$ when the subscripted condition holds and $0$ otherwise.

The fluctuation LQG data of the stationary mean-field triple $(S,A,P)$ relative to the chosen extensions and the observation-rate family consist of the following matrices, defined for each $t\in[0,T]$:

1. (State matrix.) $\mathcal{E}_t$, with $l$ rows and $l$ columns and entries $(\mathcal{E}_t)_{\gamma\delta}=\partial_\delta\bar{b}^\gamma(S_t,A_t)$ for $\gamma,\delta\in\{1,\dots,l\}$.

2. (Control matrix.) $\mathcal{B}_t$, with $l$ rows and $m$ columns and entries $(\mathcal{B}_t)_{\gamma j}=\partial_{l+j}\bar{b}^\gamma(S_t,A_t)$ for $\gamma\in\{1,\dots,l\}$ and $j\in\{1,\dots,m\}$.

3. (Observation matrix.) $\tilde{\mathcal{E}}_t$, with $\tilde{l}$ rows and $l$ columns and entries $(\tilde{\mathcal{E}}_t)_{\upsilon\gamma}=\partial_\gamma\bar{\tilde{b}}^\upsilon(S_t)$ for $\upsilon\in\{1,\dots,\tilde{l}\}$ and $\gamma\in\{1,\dots,l\}$.

4. (Hessian blocks.) $H^{SS}_t$ with $l$ rows and $l$ columns, $H^{SA}_t$ with $l$ rows and $m$ columns, $H^{AS}_t$ with $m$ rows and $l$ columns, and $H^{AA}_t$ with $m$ rows and $m$ columns, with entries

for $\gamma,\delta\in\{1,\dots,l\}$ and $j,k\in\{1,\dots,m\}$.

5. (Terminal matrix.) $F^\star$, with $l$ rows and $l$ columns and entries $(F^\star)_{\gamma\delta}=F_{\gamma\delta}$ for $\gamma,\delta\in\{1,\dots,l\}$ (independent of $t$).

6. (State noise covariance.) $\Theta^\star_t$, with $l$ rows and $l$ columns and entries $(\Theta^\star_t)_{\gamma\delta}=\Theta^{\gamma\delta}(S_t,A_t)$ for $\gamma,\delta\in\{1,\dots,l\}$.

7. (Observation noise covariance.) $\tilde{\Theta}^\star_t$, with $\tilde{l}$ rows and $\tilde{l}$ columns and entries $(\tilde{\Theta}^\star_t)_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}\,\tilde{b}^\upsilon(S_t)$ for $\upsilon,\upsilon'\in\{1,\dots,\tilde{l}\}$.