Stopping Times on a Compact Time Interval: Elementary Operations, the Prior Sigma-Algebra, Dyadic Approximation, Sampling, and Hitting Times

Let $\mathbb{R}$ be the real numbers, let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be real, and let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$. Stopping times are those of $(\mathcal{F}_t)_{t\in[0,T]}$, the $\sigma$-algebra of events prior to a stopping time $\tau$ is denoted $\mathcal{F}_\tau$, and for a family $X=(X_t)_{t\in[0,T]}$ of real-valued functions on $\Omega$ the sampled function $X_\tau$ and stopped family $X^\tau$ are as defined there. A family $X$ is adapted when $X_t$ is an $\mathcal{F}_t$-measurable random variable for every $t\in[0,T]$, progressive measurability is with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, a path of $X$ is a map $s\mapsto X_s(\omega)$ on $[0,T]$ at a fixed $\omega$, and a path is right-continuous in the sense of the progressive measurability toolkit: for every $s\in[0,T]$ and every sequence $(s_j)_{j\in\mathbb{N}}$ in $[s,T]$ converging to $s$, the sequence $(X_{s_j}(\omega))_{j\in\mathbb{N}}$ converges to $X_s(\omega)$. 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. For a natural number $n$ let

$$D_n=\{kT2^{-n}:\ k\in\{0,1,\dots,2^n\}\}$$

be the $n$-th dyadic grid of $[0,T]$. For a function $\tau:\Omega\to[0,T]$ and $t\in[0,T]$ write $\{\tau\le t\}$ for $\{\omega\in\Omega:\tau(\omega)\le t\}$, and similarly for $<$, $=$, $\ge$, $>$.

1. (Elementary stopping times.) Every constant function $\omega\mapsto c$ with $c\in[0,T]$ is a stopping time. If $\sigma$ and $\tau$ are stopping times, then so are the pointwise minimum $\min(\sigma,\tau)$ and the pointwise maximum $\max(\sigma,\tau)$. If $\tau$ is a stopping time, then for every $t\in[0,T]$ the four sets $\{\tau<t\}$, $\{\tau=t\}$, $\{\tau\ge t\}$, $\{\tau>t\}$ belong to $\mathcal{F}_t$, and $\tau$, regarded as a real-valued function on $\Omega$, is measurable with respect to $\mathcal{F}_T$ and the Borel $\sigma$-algebra of the real line.

2. (The prior $\sigma$-algebra.) Let $\sigma$ and $\tau$ be stopping times. Then $\mathcal{F}_\tau$ is a $\sigma$-algebra on $\Omega$ contained in $\mathcal{F}_T$; the function $\tau$ is measurable with respect to $\mathcal{F}_\tau$ and the Borel $\sigma$-algebra; if $\tau$ is the constant $c\in[0,T]$ then $\mathcal{F}_\tau=\mathcal{F}_c$; if $\sigma(\omega)\le\tau(\omega)$ for every $\omega\in\Omega$, then $\mathcal{F}_\sigma\subseteq\mathcal{F}_\tau$; and for every $s\in[0,T]$ and every $A\in\mathcal{F}_s$,

$$A\cap\{\tau>s\}\in\mathcal{F}_{\min(s,\tau)},$$

where $\min(s,\tau)$ denotes the stopping time $\omega\mapsto\min(s,\tau(\omega))$.

3. (Dyadic approximation from above.) Let $\tau$ be a stopping time and, for each natural number $n$, define

$$\tau_n(\omega)=\min\{d\in D_n:\ d\ge\tau(\omega)\}\qquad(\omega\in\Omega),$$

the least grid point of $D_n$ not below $\tau(\omega)$ (which exists since $T\in D_n$). Then $\tau_n$ is a stopping time with values in $D_n$; for every $\omega$ and $n$,

$$\tau(\omega)\le\tau_{n+1}(\omega)\le\tau_n(\omega)\le\tau(\omega)+T2^{-n},$$

so that $(\tau_n(\omega))_{n\in\mathbb{N}}$ converges to $\tau(\omega)$ for every $\omega\in\Omega$; $\{\tau_n=0\}=\{\tau=0\}$ and, for $k\in\{1,\dots,2^n\}$, $\{\tau_n=kT2^{-n}\}=\{(k-1)T2^{-n}<\tau\le kT2^{-n}\}$; $\mathcal{F}_\tau\subseteq\mathcal{F}_{\tau_n}$; and if $\sigma$ is a stopping time with $\sigma\le\tau$ pointwise, then $\sigma_n\le\tau_n$ pointwise, where $\sigma_n$ is defined from $\sigma$ in the same way.

4. (Sampling a progressively measurable process.) Let $X=(X_t)_{t\in[0,T]}$ be progressively measurable and let $\tau$ be a stopping time.

(i) For every $t\in[0,T]$ the function $X^\tau_t=X_{\min(t,\tau)}$ is $\mathcal{F}_t$-measurable; thus the stopped family $X^\tau$ is adapted.

(ii) The sampled function $X_\tau$ is measurable with respect to $\mathcal{F}_\tau$ and the Borel $\sigma$-algebra; in particular it is an $\mathcal{F}_T$-measurable random variable.

(iii) Let $\omega\in\Omega$ be a point at which the path of $X$ is right-continuous, and let $\tau_n$ be as in claim 3. Then the sequence $(X_{\tau_n}(\omega))_{n\in\mathbb{N}}$ converges to $X_\tau(\omega)$, and the path of $X^\tau$ at $\omega$ is right-continuous. Consequently, if $X$ is adapted and every path of $X$ is right-continuous, then $X^\tau$ is adapted with every path right-continuous, hence progressively measurable.

5. (Hitting times of continuous adapted processes.) Let $Y=(Y_t)_{t\in[0,T]}$ be adapted with every path continuous on $[0,T]$, and let $c$ be a real number. For $\omega\in\Omega$ let $H(\omega)=\{t\in[0,T]:Y_t(\omega)\ge c\}$ and define

$$\tau(\omega)=\begin{cases}\inf H(\omega)&\text{if }H(\omega)\neq\emptyset,\\ T&\text{if }H(\omega)=\emptyset,\end{cases}$$

the infimum in the first case being the greatest lower bound, which exists by the existence theorem for infima since there $H(\omega)$ is nonempty and bounded below by $0$. Then $\tau$ is a stopping time; $Y_t(\omega)<c$ for every $\omega$ and every $t\in[0,T]$ with $t<\tau(\omega)$; and $Y_{\tau(\omega)}(\omega)\ge c$ whenever $H(\omega)\neq\emptyset$, in particular for every $\omega$ with $\tau(\omega)<T$.