Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow

Adopt the setting and notation of the flow stability lemma: an affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$, its transition-rate family $\beta$ with rate bound $B$, state-Lipschitz constant $\Lambda_b$ and projected drift $\hat{b}$; a horizon $T>0$; the constant $K_b=2\sqrt{l}\,(l-1)B$; the probability simplex $\Delta^l$; the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, its admissible representatives, and the mean-field flow $S(x_0,\xi)$ of claim 2 of that lemma, with values $S_t(x_0,\xi)\in\Delta^l$; and the metric $\rho$ on $\mathcal{U}_{\mathcal{A}}$ of claim 1 of the weak metrizability and compactness theorem, formed from a fixed dense sequence, with Borel $\sigma$-algebra $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$. Write $|\cdot|$ for the Euclidean norm, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, and, for $0<s\le T$, $\mathcal{B}_{[0,s]}$, $\lambda_{[0,s]}$ for the trace Borel $\sigma$-algebra and restricted Lebesgue measure on $[0,s]$, both restrictions of Lebesgue measure on the real line by claim 1 of that toolkit. The constant $K_1$ and the bound $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ are those of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, adopted through the flow stability lemma. Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

1. (Causality of the flow.) Let $x_0\in\Delta^l$, let $t\in[0,T]$, and let $\xi,\zeta\in\mathcal{U}_{\mathcal{A}}$ have admissible representatives $u$ and $v$ with $\lambda_{[0,T]}\bigl(\{s\in[0,t]:u(s)\neq v(s)\}\bigr)=0$, the set in question belonging to $\mathcal{B}_{[0,T]}$ since $s\mapsto|u(s)-v(s)|$ is measurable (its components by The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval, the norm of a measurable map by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable) and $[0,t]$ is a Borel set. Then $S_s(x_0,\xi)=S_s(x_0,\zeta)$ for every $s\in[0,t]$.

2. (Borel measurability in the control.) Let $x_0\in\Delta^l$, $t\in[0,T]$ and $\gamma\in\{1,\dots,l\}$. Then the map $\xi\mapsto S^\gamma_t(x_0,\xi)$ from $\mathcal{U}_{\mathcal{A}}$ to $\mathbb{R}$ is measurable with respect to $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ and $\mathcal{B}(\mathbb{R})$.

For claims 3 and 4 adopt in addition the setting and notation of the progressive measurability lemma for the realized control, with the same control set $\mathcal{A}$ (nonempty, convex and compact as part of the affine-controlled data), the same rate family $\beta$, the same horizon $T$, the same metric $\rho$, and with the bound $R$ there taken to be $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data: the solution of the controlled $N$-agent dynamics on the $N$-agent driving system $(\Omega,\mathcal{F},P)$ for an $\mathcal{A}$-valued observation-driven control policy, with observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$; the realized control $\hat{\alpha}$ of the realized-control lemma, with $\hat{\alpha}(\omega)\in\mathcal{U}_{\mathcal{A}}$ for every $\omega\in\Omega$; and the notions of a $\mathcal{G}_t$-measurable function and of progressive measurability fixed there. Fix $x_0\in\Delta^l$ and define the realized mean-field flow $\Phi=(\Phi_t)_{t\in[0,T]}$ by

with components $\Phi^\gamma_t(\omega)$.

3. (The realized flow is adapted, Lipschitz, and progressively measurable.) $\Phi_0(\omega)=x_0$ and $|\Phi_t(\omega)-\Phi_r(\omega)|\le K_b|t-r|$ for all $r,t\in[0,T]$ and every $\omega\in\Omega$; for every $t\in[0,T]$ and every $\gamma$ the function $\Phi^\gamma_t$ is $\mathcal{G}_t$-measurable; every path $t\mapsto\Phi^\gamma_t(\omega)$ is continuous on $[0,T]$; and $\Phi$ is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$, hence also with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$.

4. (The deviation process.) Let $S^*:[0,T]\to\mathbb{R}^l$ be a map each of whose components $t\mapsto S^{*\gamma}_t$ is continuous on $[0,T]$, and define $Y_t(\omega)=|\Phi_t(\omega)-S^*_t|$ for $t\in[0,T]$ and $\omega\in\Omega$. Then $Y_t$ is $\mathcal{G}_t$-measurable for every $t$, every path $t\mapsto Y_t(\omega)$ is continuous on $[0,T]$, and there is a real $K_Y\ge0$ with $0\le Y_t(\omega)\le K_Y$ for all $t$ and $\omega$. Moreover $\overline{Y}(\omega)=\sup_{t\in[0,T]}Y_t(\omega)$ exists for every $\omega\in\Omega$, and $\overline{Y}$ is $\mathcal{G}_T$-measurable.