Mean-Square Linearization Residual of the Observation Fluctuation Process

Adopt the setting of the fluctuation processes of the controlled $N$-agent dynamics: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, an observation-rate family $\tilde{\beta}$ with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, observation processes $\Upsilon^\upsilon_t$, and control $\alpha_t$, a mean-field trajectory pair $(S,A)$ for $\beta$ with horizon $T$, and the state fluctuation process $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$. Assume that $\tilde{\beta}$ admits a twice continuously differentiable extension $(\tilde{U},\bar{\tilde{\beta}})$ with derivative bound $\tilde{K}$. Let $\tilde{b}$ be the aggregate observation drift of $\tilde{\beta}$, with $\tilde{b}(\Sigma)$ the vector with components $\tilde{b}^1(\Sigma),\dots,\tilde{b}^{\tilde{l}}(\Sigma)$, let $\bar{\tilde{b}}$ be the extended aggregate observation drift of $(\tilde{U},\bar{\tilde{\beta}})$, adopt the partial-derivative notation $\partial_\gamma$, $\partial_\delta\partial_\gamma$ of the extension definition, and let $\tilde{M}_t=(\tilde{M}^\upsilon_t)_{\upsilon\in\{1,\dots,\tilde{l}\}}$ be as in the martingale decomposition. Write $\mathbb{E}$ for the expectation, $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), $\Delta^l$ for the probability simplex, and $\mathbf{1}_{D}$ for the function equal to $1$ on a set $D$ and $0$ off $D$, and set, for $s\in[0,T]$,

Define the observation linearization matrix: for $s\in[0,T]$, the real matrix $\tilde{\mathcal{E}}_s$ with $\tilde{l}$ rows and $l$ columns and entries

which is defined because $S_s$ lies in $\Delta^l\subseteq\tilde{U}$ (the inclusion being part of the extension definition) and the partial derivatives of $\bar{\tilde{b}}$ exist there by the regularity of the extended aggregate observation drift; when $(S,A)$ belongs to a stationary mean-field triple and the observation-rate extension is the one chosen there, $\tilde{\mathcal{E}}_s$ is the observation matrix of the fluctuation LQG data, with the same symbol, by the identical defining formula of clause 3 of that definition. Define the observation linearization residual, with the matrix-vector product $\tilde{\mathcal{E}}_s\mathfrak{s}_s$,

and the observation fluctuation process, with $\Upsilon_t$ the vector with components $\Upsilon^1_t,\dots,\Upsilon^{\tilde{l}}_t$,

the integral being the componentwise Lebesgue integral over the compact interval $[0,t]$, equal to $0$ for $t=0$. Here each $r\mapsto\tilde{b}^\upsilon(S_r)$ is continuous on $[0,T]$ — relative to $[0,T]$, both $[0,T]$ and the codomain $\mathbb{R}$ carrying the metric of the real line — because the vector map $r\mapsto S_r$ is continuous into $\mathbb{R}^l$ in the Euclidean sense ($d(S_r,S_{r'})\le\sum_{\gamma=1}^{l}|S^\gamma_r-S^\gamma_{r'}|$ by claim 1 of the componentwise estimates, each summand controlled by the componentwise continuity of the trajectory pair), $\tilde{b}^\upsilon$ agrees on $\Delta^l$ with the extension $\bar{\tilde{b}}^\upsilon$, which is continuous in the Euclidean sense by part (i) of the regularity lemma, compositions preserve continuity by the composition theorem, and the Euclidean and metric notions agree for real-valued maps by claim 1 of the continuity agreement lemma; for $t>0$ the restriction to $[0,t]$ is continuous by claim 1 of restriction stability and its Lebesgue integral exists by claim 3 of the toolkit. Then:

(a) (Pointwise bounds.) At every point of $[0,T]\times\Omega$,

moreover $|\mathfrak{s}_s|\le2\sqrt{N}$ at every point of $[0,T]\times\Omega$, so that $|\tilde{e}_s|\le4\,\tilde{\Lambda}\,\sqrt{N}$ everywhere.

(b) (Well-definedness.) Each map $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\tilde{e}^\upsilon_s(\omega)$ is measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$, by the joint measurability of the state and control together with measurability of sequentially continuous functions of measurable Euclidean maps and the continuity of the mean-field trajectory pair; and at each $\omega\in\Omega_0$ every section $s\mapsto\tilde{e}^\upsilon_s(\omega)$ is bounded and Lebesgue integrable over $[0,t]$ for every $t\in[0,T]$. Define the observation residual process

Each $\tilde{\mathcal{R}}^\upsilon_t$ is a random variable, by the Tonelli theorem applied to the positive and negative parts of $\mathbf{1}_{\Omega_0}\tilde{e}^\upsilon$.

(c) (Integral form of the observation fluctuation process.) At every $\omega\in\Omega_0$ — an event of probability $1$, on which clause (a) of the martingale decomposition applies — and hence almost surely, for every $t\in[0,T]$,

all integrals existing componentwise at such $\omega$.

(d) (Mean-square residual bound.) For every $t\in[0,T]$, all quantities below are finite and

Here the maps $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\min(\cdots)^2$ and $(s,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,|\mathfrak{s}_s(\omega)|^4$ are product-measurable, being continuous functions of product-measurable maps furnished by the joint measurability lemma, so the integrands $s\mapsto\mathbb{E}[\min(\cdots)^2]$ and $s\mapsto\mathbb{E}[|\mathfrak{s}_s|^4]$ (unchanged by the $\mathbf{1}_{\Omega_0}$ modification, $\Omega_0$ having probability $1$) are measurable functions of $s$ by the Tonelli theorem, the trace Lebesgue measure and $P$ being finite, hence $\sigma$-finite, measures. In particular