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

Claim 1. By claim 4 of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow, the family $Y=(Y_t)_{t\in[0,T]}$ is adapted to $(\mathcal{G}_t)_{t\in[0,T]}$ with every path continuous on $[0,T]$; by claim 5 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration, so is $\mathcal{E}=(\mathcal{E}_t)_{t\in[0,T]}$. Hence $\tau_Y$ and $\tau_{\mathcal{E}}$, which are the hitting times of claim 5 of Stopping Times on a Compact Time Interval: Elementary Operations, the Prior Sigma-Algebra, Dyadic Approximation, Sampling, and Hitting Times for $Y$ with level $c=\varepsilon_Y$ and for $\mathcal{E}$ with level $c=c_{\mathcal{E}}$, are stopping times of $(\mathcal{G}_t)_{t\in[0,T]}$, and so is their pointwise minimum $\tau$ by claim 1 of the same toolkit. Since $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$ for every $t$ (claim 1 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration), the defining condition $\{\theta\le t\}\in\mathcal{G}_t$ of a stopping time implies $\{\theta\le t\}\in\mathcal{F}^{\mathrm{sys}}_t$, so all three are stopping times of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ as well. The membership $\{\tau\le t\}\in\mathcal{G}_t$ is the definition, $\{t<\tau\}=\{\tau>t\}\in\mathcal{G}_t$ is claim 1 of the toolkit, and $\{\tau<T\}\in\mathcal{G}_T$ is the same claim with $t=T$.

Claim 2. Let $\omega\in\Omega$ and $t<\tau(\omega)$. Then $t<\tau_Y(\omega)$ and $t<\tau_{\mathcal{E}}(\omega)$, so claim 5 of Stopping Times on a Compact Time Interval: Elementary Operations, the Prior Sigma-Algebra, Dyadic Approximation, Sampling, and Hitting Times gives $Y_t(\omega)<\varepsilon_Y$ and $\mathcal{E}_t(\omega)<c_{\mathcal{E}}$.

For the bounds at the stopping time we first show: if $\mathcal{E}_0(\omega)<c_{\mathcal{E}}$ then $\mathcal{E}_{\tau_{\mathcal{E}}(\omega)}(\omega)\le c_{\mathcal{E}}$. If $H_{\mathcal{E}}(\omega)=\emptyset$ then $\tau_{\mathcal{E}}(\omega)=T$ and $\mathcal{E}_T(\omega)<c_{\mathcal{E}}$. Otherwise $\theta=\tau_{\mathcal{E}}(\omega)$ is the greatest lower bound of $H_{\mathcal{E}}(\omega)$, so no $u<\theta$ lies in $H_{\mathcal{E}}(\omega)$, that is, $\mathcal{E}_u(\omega)<c_{\mathcal{E}}$ for every $u\in[0,\theta)$. Suppose $\mathcal{E}_\theta(\omega)>c_{\mathcal{E}}$. Then $\theta>0$, since $\mathcal{E}_0(\omega)<c_{\mathcal{E}}$; by continuity of the path at $\theta$ relative to $[0,T]$, applied with $\varepsilon=\mathcal{E}_\theta(\omega)-c_{\mathcal{E}}>0$, there is $\delta>0$ with $\mathcal{E}_u(\omega)>c_{\mathcal{E}}$ for every $u\in[0,T]$ with $|u-\theta|<\delta$, and any $u\in(\max(0,\theta-\delta),\theta)$ gives a contradiction. Hence $\mathcal{E}_\theta(\omega)\le c_{\mathcal{E}}$. The same argument with $Y$, $\varepsilon_Y$, $H_Y(\omega)$ and $\tau_Y(\omega)$ in place of $\mathcal{E}$, $c_{\mathcal{E}}$, $H_{\mathcal{E}}(\omega)$ and $\tau_{\mathcal{E}}(\omega)$ shows: if $Y_0(\omega)<\varepsilon_Y$ then $Y_{\tau_Y(\omega)}(\omega)\le\varepsilon_Y$.

Now $\mathcal{E}_0(\omega)=0<c_{\mathcal{E}}$ by the definition of $\mathcal{E}$ (the integral over $[0,0]$ being $0$). Let $t\in[0,T]$ and $s=\min(t,\tau(\omega))$. If $s<\tau_{\mathcal{E}}(\omega)$ then $\mathcal{E}_s(\omega)<c_{\mathcal{E}}$ by claim 5 of the toolkit; otherwise $s=\tau(\omega)=\tau_{\mathcal{E}}(\omega)$, since $s\le\tau(\omega)\le\tau_{\mathcal{E}}(\omega)$, and $\mathcal{E}_s(\omega)\le c_{\mathcal{E}}$ by the display just proved. If moreover $|x_0-S^*_0|<\varepsilon_Y$, then $Y_0(\omega)=|\Phi_0(\omega)-S^*_0|=|x_0-S^*_0|<\varepsilon_Y$ by claim 3 of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow, and the same dichotomy with $\tau_Y$ in place of $\tau_{\mathcal{E}}$ gives $Y_s(\omega)\le\varepsilon_Y$.

Claim 3. Since $\tau=\min(\tau_Y,\tau_{\mathcal{E}})$ pointwise, $\tau(\omega)<T$ holds exactly when $\tau_Y(\omega)<T$ or $\tau_{\mathcal{E}}(\omega)<T$, which is the stated identity. If $\tau_Y(\omega)<T$ then $H_Y(\omega)\neq\emptyset$, so some $t\in[0,T]$ has $Y_t(\omega)\ge\varepsilon_Y$, and $\overline{Y}(\omega)\ge Y_t(\omega)\ge\varepsilon_Y$, the supremum existing by claim 4 of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow. If $\tau_{\mathcal{E}}(\omega)<T$ then $H_{\mathcal{E}}(\omega)\neq\emptyset$, so some $t$ has $\mathcal{E}_t(\omega)\ge c_{\mathcal{E}}$, and $\mathcal{E}_T(\omega)\ge\mathcal{E}_t(\omega)\ge c_{\mathcal{E}}$ because the paths of $\mathcal{E}$ are nondecreasing by claim 5 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration. If $\tau_Y(\omega)=T$, then either $H_Y(\omega)=\emptyset$, in which case $Y_t(\omega)<\varepsilon_Y$ for every $t$, or $H_Y(\omega)\neq\emptyset$ with greatest lower bound $T$, in which case $H_Y(\omega)=\{T\}$ (every element of $H_Y(\omega)\subseteq[0,T]$ being at least its greatest lower bound $T$) and $Y_t(\omega)<\varepsilon_Y$ for $t<T$ while $Y_T(\omega)\ge\varepsilon_Y$; in the second case, continuity of the path at $T$ relative to $[0,T]$ forces $Y_T(\omega)\le\varepsilon_Y$, for if $Y_T(\omega)>\varepsilon_Y$ there would be $\delta>0$ with $Y_t(\omega)>\varepsilon_Y$ for all $t\in(T-\delta,T]$, contradicting $Y_t(\omega)<\varepsilon_Y$ for $t<T$. In both cases $Y_t(\omega)\le\varepsilon_Y$ for every $t$, so $\overline{Y}(\omega)\le\varepsilon_Y$.

The three sets on the right lie in $\mathcal{G}_T$ because $\overline{Y}$ is $\mathcal{G}_T$-measurable (claim 4 of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow) and $\mathcal{E}_T$ is $\mathcal{G}_T$-measurable (claim 5 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration), indeed $\{\overline{Y}\le\varepsilon_Y\}$ is the complement of $\{\overline{Y}>\varepsilon_Y\}\in\mathcal{G}_T$, and $\{\overline{Y}\ge\varepsilon_Y\}=\bigcap_{n\ge1}\{\overline{Y}>\varepsilon_Y-1/n\}$ with every member in $\mathcal{G}_T$, both by the criterion of claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line; likewise for $\{\mathcal{E}_T\ge c_{\mathcal{E}}\}$. Finally $\{\tau<T\}\subseteq\{\overline{Y}\ge\varepsilon_Y\}\cup\{\mathcal{E}_T\ge c_{\mathcal{E}}\}$ by the inclusions just proved, and the probability of a union of two events is at most the sum of their probabilities, by monotonicity (claim 2) and countable subadditivity (claim 4, the sequence padded with the empty set) of Basic Properties of a Measure; this is the final display. $\blacksquare$