The Good-Set Stopping Time of the Realized Control: Flow Deviation and Control Energy

Adopt the setting and notation of claims 3 and 4 of the adaptedness lemma for the realized mean-field flow, together with those of the progressive measurability lemma for the realized control on which it rests: the affine-controlled transition-rate family with compact convex control set $\mathcal{A}$ and its transition-rate family $\beta$; the horizon $T>0$; the solution of the controlled $N$-agent dynamics on the $N$-agent driving system $(\Omega,\mathcal{F},P)$ 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 $\hat{\alpha}$; a point $x_0$ of the probability simplex $\Delta^l$ and the realized mean-field flow $\Phi_t(\omega)=S_t(x_0,\hat{\alpha}(\omega))$; a map $S^*:[0,T]\to\mathbb{R}^l$ with continuous components and the deviation process $Y_t(\omega)=|\Phi_t(\omega)-S^*_t|$ with supremum $\overline{Y}$, where $|\cdot|$ is the Euclidean norm; a map $A:[0,T]\to\mathcal{A}$ each of whose components is measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra of the real line, and the control energy process $\mathcal{E}_t(\omega)=\int_{[0,t]}|\hat{\alpha}(s,\omega)-A_s|^2\,ds$ of claim 5 of the progressive measurability lemma. Stopping times are those of whichever of the two filtrations $(\mathcal{G}_t)_{t\in[0,T]}$ and $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ is named, with time index restricted to $[0,T]$, and for a function $\theta:\Omega\to[0,T]$ and $t\in[0,T]$ we write $\{t<\theta\}$ for $\{\omega\in\Omega:t<\theta(\omega)\}$, and similarly for the other order relations. 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.

Fix a real number $\varepsilon_Y$ and a real number $c_{\mathcal{E}}>0$. For $\omega\in\Omega$ put

$$H_Y(\omega)=\{t\in[0,T]:Y_t(\omega)\ge\varepsilon_Y\},\qquad H_{\mathcal{E}}(\omega)=\{t\in[0,T]:\mathcal{E}_t(\omega)\ge c_{\mathcal{E}}\},$$

and define $\tau_Y(\omega)$ to be the greatest lower bound of $H_Y(\omega)$ if $H_Y(\omega)\neq\emptyset$ and $T$ otherwise, $\tau_{\mathcal{E}}(\omega)$ to be the greatest lower bound of $H_{\mathcal{E}}(\omega)$ if $H_{\mathcal{E}}(\omega)\neq\emptyset$ and $T$ otherwise (in each case the greatest lower bound exists by the existence theorem for infima, the set being nonempty in the case at hand and bounded below by $0$), and the good-set time (a stopping time by claim 1 below)

$$\tau(\omega)=\min\bigl(\tau_Y(\omega),\tau_{\mathcal{E}}(\omega)\bigr).$$

Then the following hold.

1. (Stopping times.) Each of $\tau_Y$, $\tau_{\mathcal{E}}$ and $\tau$ is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$ and of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$. In particular $\{t<\tau\}\in\mathcal{G}_t$ and $\{\tau\le t\}\in\mathcal{G}_t$ for every $t\in[0,T]$, and $\{\tau<T\}\in\mathcal{G}_T$.

2. (Bounds before and at the stopping time.) For every $\omega\in\Omega$ and every $t\in[0,T]$ with $t<\tau(\omega)$,

$$Y_t(\omega)<\varepsilon_Y\qquad\text{and}\qquad\mathcal{E}_t(\omega)<c_{\mathcal{E}} .$$

Moreover $\mathcal{E}_{\min(t,\tau(\omega))}(\omega)\le c_{\mathcal{E}}$ for every $\omega\in\Omega$ and every $t\in[0,T]$; and if $|x_0-S^*_0|<\varepsilon_Y$, then also $Y_{\min(t,\tau(\omega))}(\omega)\le\varepsilon_Y$ for every $\omega\in\Omega$ and every $t\in[0,T]$.

3. (Early stopping.) $\{\tau<T\}=\{\tau_Y<T\}\cup\{\tau_{\mathcal{E}}<T\}$, with

$$\{\tau_Y<T\}\subseteq\{\overline{Y}\ge\varepsilon_Y\},\qquad\{\tau_{\mathcal{E}}<T\}\subseteq\{\mathcal{E}_T\ge c_{\mathcal{E}}\},\qquad\{\tau_Y=T\}\subseteq\{\overline{Y}\le\varepsilon_Y\},$$

all three sets on the right belonging to $\mathcal{G}_T$; consequently

$$P(\tau<T)\le P\bigl(\overline{Y}\ge\varepsilon_Y\bigr)+P\bigl(\mathcal{E}_T\ge c_{\mathcal{E}}\bigr).$$