The Extended Good-Set Stopping Time of the Realized Control: Clipped-Out Time and Energy Hitting Bounds

Adopt the setting and notation of the good-set stopping-time lemma, together with those of the progressive measurability lemma for the realized control and of the causality and adaptedness lemma for the realized mean-field flow on which it rests: the affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^m$; its transition-rate family $\beta$, together with the constants $R=\sup_{a\in\mathcal{A}}|a|$, $K_1$, the rate bound $B$ and the state-Lipschitz constant $\Lambda_b$ of that lemma, its aggregate state drift $b$, and the constant $K_2=2\sqrt{l}\,(l-1)K_1$ fixed in the preamble of the flow stability lemma and used in claim 1 there; the horizon $T>0$; 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, each of whose paths $s\mapsto\hat{\alpha}(s,\omega)$ takes all its values in $\mathcal{A}$; the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, with the convention of claim 5 of the definition of the Lebesgue space of square-integrable vector-valued maps by which an element is denoted by the same symbol as a representative of it, and the admissible representatives and the mean-field flow of claim 2 of the flow stability lemma, written $S(z_0,\xi)$ here for an initial state $z_0\in\Delta^l$ and a control $\xi\in\mathcal{U}_{\mathcal{A}}$; the 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 $\mathcal{B}_{[0,T]}$ 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; and real numbers $\varepsilon_Y$ and $c_{\mathcal{E}}>0$ with the times $\tau_Y$, $\tau_{\mathcal{E}}$ and the good-set time $\tau=\min(\tau_Y,\tau_{\mathcal{E}})$ of the good-set stopping-time 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 $[0,T]$; 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; $\int_{[0,t]}\cdot\,ds$ denotes the Lebesgue integral over the compact interval $[0,t]$, taken to be $0$ for $t=0$; $\mathbb{E}$ is the expectation; $\exp$ is the exponential function; and $\sqrt{\cdot}$ is the nonnegative square root. 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 real numbers $\delta>0$ and $\theta_{\mathrm{out}}>0$. Define $I^{\mathrm{out}}:[0,T]\times\Omega\to\mathbb{R}$ by

$$I^{\mathrm{out}}_s(\omega)=1\ \text{ if }\ |\hat{\alpha}(s,\omega)-A_s|>\delta,\qquad I^{\mathrm{out}}_s(\omega)=0\ \text{ otherwise},$$

and the clipped-out time process

$$\mathcal{O}_t(\omega)=\int_{[0,t]}I^{\mathrm{out}}_s(\omega)\,ds\qquad(t\in[0,T],\ \omega\in\Omega).$$

For $\omega\in\Omega$ put $H_{\mathcal{O}}(\omega)=\{t\in[0,T]:\mathcal{O}_t(\omega)\ge\theta_{\mathrm{out}}\}$, define $\tau_{\mathrm{out}}(\omega)$ to be the greatest lower bound of $H_{\mathcal{O}}(\omega)$ if $H_{\mathcal{O}}(\omega)\neq\emptyset$ and $T$ otherwise (the greatest lower bound existing by the existence theorem for infima, the set being nonempty in the case at hand and bounded below by $0$), and define the extended good-set time

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

Set $C_S=e^{\Lambda_bT}\sqrt{l}\,K_2\sqrt{T}$.

Then the following hold.

1. (The clipped-out time process.) The family $(I^{\mathrm{out}}_s)_{s\in[0,T]}$ is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$, with values in $\{0,1\}$. The family $\mathcal{O}=(\mathcal{O}_t)_{t\in[0,T]}$ is progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$, hence also with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$; each $\mathcal{O}_t$ is $\mathcal{G}_t$-measurable; and for all $0\le t_0\le t\le T$ and every $\omega\in\Omega$,

$$0\le\mathcal{O}_t(\omega)-\mathcal{O}_{t_0}(\omega)\le t-t_0 ,$$

so that every path $t\mapsto\mathcal{O}_t(\omega)$ is nondecreasing and continuous on $[0,T]$ with $\mathcal{O}_0(\omega)=0$ and $\mathcal{O}_T(\omega)\le T$. Moreover

$$\delta^2\,\mathcal{O}_t(\omega)\le\mathcal{E}_t(\omega)\qquad\text{for every }t\in[0,T]\text{ and every }\omega\in\Omega .$$

2. (Stopping and hitting.) $\tau_{\mathrm{out}}$ is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$ and of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and $\{\tau_{\mathrm{out}}\le q\}=\{\mathcal{O}_q\ge\theta_{\mathrm{out}}\}$ for every $q\in[0,T)$. For every $\omega\in\Omega$ and every $t\in[0,T]$ with $t<\tau_{\mathrm{out}}(\omega)$ one has $\mathcal{O}_t(\omega)<\theta_{\mathrm{out}}$; furthermore $\mathcal{O}_{\min(t,\tau_{\mathrm{out}}(\omega))}(\omega)\le\theta_{\mathrm{out}}$ for every $t\in[0,T]$ and $\omega\in\Omega$; and $\mathcal{O}_{\tau_{\mathrm{out}}(\omega)}(\omega)\ge\theta_{\mathrm{out}}$ whenever $\tau_{\mathrm{out}}(\omega)<T$, so that $\{\tau_{\mathrm{out}}<T\}\subseteq\{\mathcal{O}_T\ge\theta_{\mathrm{out}}\}$.

3. (The extended good-set time.) $\tau^*$ is a stopping time of $(\mathcal{G}_t)_{t\in[0,T]}$ and of $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and $\{t<\tau^*\}\in\mathcal{G}_t$ for every $t\in[0,T]$. For every $\omega\in\Omega$ and every $t\in[0,T]$ with $t<\tau^*(\omega)$,

$$Y_t(\omega)<\varepsilon_Y,\qquad\mathcal{E}_t(\omega)<c_{\mathcal{E}},\qquad\mathcal{O}_t(\omega)<\theta_{\mathrm{out}} .$$

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

$$P(\tau^*<T)\le P\bigl(\overline{Y}\ge\varepsilon_Y\bigr)+P\bigl(\mathcal{E}_T\ge c_{\mathcal{E}}\bigr)+P\bigl(\mathcal{O}_T\ge\theta_{\mathrm{out}}\bigr),$$

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

4. (Deviation-energy bound.) The path $A$ is an admissible representative of an element of $\mathcal{U}_{\mathcal{A}}$, written $\zeta_A$ (its components are measurable, every value lies in $\mathcal{A}$, and it is bounded by $R$, hence square-integrable). Suppose that $S^*$ takes values in $\Delta^l$, that $S^*_0=x_0$, and that

$$S^{*\gamma}_t=x^\gamma_0+\int_{[0,t]}b^\gamma(S^*_s,A_s)\,ds\qquad\text{for all }t\in[0,T]\text{ and }\gamma\in\{1,\dots,l\}.$$

Then $S^*_t=S_t(x_0,\zeta_A)$ for every $t\in[0,T]$, and for every $\omega\in\Omega$ and every $t\in[0,T]$,

$$Y_t(\omega)\ \le\ e^{\Lambda_bT}\sqrt{l}\,K_2\int_{[0,T]}\mathbf{1}_{[0,t]}(s)\,\bigl|\hat{\alpha}(s,\omega)-A_s\bigr|\,ds\ \le\ C_S\,\bigl(\mathcal{E}_t(\omega)\bigr)^{1/2},$$

where $\mathbf{1}_{[0,t]}$ is the function on $[0,T]$ equal to $1$ on $[0,t]$ and $0$ elsewhere.

5. (Escape probability.) Assume the hypotheses of claim 4 and $\varepsilon_Y>0$. Set $m^*=\min\bigl(\varepsilon_Y^2/C_S^2,\ c_{\mathcal{E}},\ \delta^2\theta_{\mathrm{out}}\bigr)$ if $C_S>0$ and $m^*=\min\bigl(c_{\mathcal{E}},\ \delta^2\theta_{\mathrm{out}}\bigr)$ if $C_S=0$; then $m^*>0$. The function $\omega\mapsto\mathcal{E}_{\tau^*(\omega)}(\omega)$ is a random variable with values in $[0,4R^2T]$,

$$\{\tau^*<T\}\subseteq\bigl\{\mathcal{E}_{\tau^*}\ge m^*\bigr\},\qquad\text{and}\qquad P(\tau^*<T)\le\frac{\mathbb{E}\bigl[\mathcal{E}_{\tau^*}\bigr]}{m^*} .$$