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 $s\in[0,T]$,

$$\tilde{g}_s=\sqrt{N}\,\big(\tilde{b}(\Sigma_s)-\tilde{b}(S_s)\big)\in\mathbb{R}^{\tilde{l}},\qquad \tilde{\Lambda}=\sqrt{l\,\tilde{l}}\;(\tilde{B}+\tilde{K}).$$

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

$$(\tilde{\mathcal{E}}_s)_{\upsilon\gamma}=\partial_\gamma\bar{\tilde{b}}^\upsilon(S_s)\qquad(\upsilon\in\{1,\dots,\tilde{l}\},\ \gamma\in\{1,\dots,l\}),$$

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 $\tilde{\mathcal{E}}_s\mathfrak{s}_s$,

$$\tilde{e}_s=\tilde{g}_s-\tilde{\mathcal{E}}_s\mathfrak{s}_s\in\mathbb{R}^{\tilde{l}}\qquad(s\in[0,T]),$$

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

$$\mathfrak{u}_t=\sqrt{N}\,\Big(\Upsilon_t-\int_{[0,t]}\tilde{b}(S_r)\,dr\Big)\in\mathbb{R}^{\tilde{l}}\qquad(t\in[0,T]),$$

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 $[0,T]\times\Omega$,

$$|\tilde{e}_s|\ \le\ \frac{3\,l\,\sqrt{\tilde{l}}\,\tilde{K}}{2\,\sqrt{N}}\,|\mathfrak{s}_s|^2\qquad\text{and}\qquad |\tilde{e}_s|\ \le\ 2\,\tilde{\Lambda}\,|\mathfrak{s}_s|;$$

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

$$\tilde{\mathcal{R}}_t=\int_{[0,t]}\tilde{e}_s\,ds\ \text{ componentwise at each }\omega\in\Omega_0\ \text{ (the integral being $0$ for $t=0$)},\qquad \tilde{\mathcal{R}}_t=0\ \text{ off }\Omega_0\qquad(t\in[0,T]).$$

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 $t\in[0,T]$,

$$\mathfrak{u}_t=\int_{[0,t]}\big(\tilde{\mathcal{E}}_r\,\mathfrak{s}_r+\tilde{e}_r\big)\,dr+\sqrt{N}\,\tilde{M}_t,$$

all integrals existing componentwise at such $\omega$.

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

$$\mathbb{E}\big[|\tilde{\mathcal{R}}_t|^2\big]\ \le\ t\int_{[0,t]}\mathbb{E}\Big[\min\Big(\frac{3\,l\,\sqrt{\tilde{l}}\,\tilde{K}}{2\,\sqrt{N}}\,|\mathfrak{s}_s|^2,\ 2\,\tilde{\Lambda}\,|\mathfrak{s}_s|\Big)^{\!2}\Big]\,ds .$$

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

$$\mathbb{E}\big[|\tilde{\mathcal{R}}_t|^2\big]\ \le\ \frac{9\,l^2\,\tilde{l}\,\tilde{K}^2}{4\,N}\;t\int_{[0,t]}\mathbb{E}\big[|\mathfrak{s}_s|^4\big]\,ds .$$