Stopping times: definition, measurability and closure properties

Give an example of a random variable $T$ and two filtrations $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$ and $\left\{ \mathcal{G}_{n}\right\}_{n \geq 1}$ such that $T$ is a stopping time w.r.t. the filtration $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$ but not w.r.t. $\left\{ \mathcal{G}_{n}\right\}_{n \geq 1}$.

Let $T_{1}$ and $T_{2}$ be stopping times w.r.t. a filtration $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$. Verify that $\min \left(T_{1}, T_{2}\right), \max \left(T_{1}, T_{2}\right), T_{1}+T_{2}$, and $T_{1}^{2}$ are stopping times w.r.t. $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$. Give an example to show that $\sqrt{T_{1}}$ and $T_{1}-1$ need not be stopping times w.r.t. $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$.

Let $T$ be a random variable taking values in $\left\{ 1,2,3, \ldots \right\}$. Show that there is a filtration $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$ w.r.t. which $T$ is a stopping time.

Let $\left\{ \mathcal{F}_{n}\right\}_{n \geq 1}$ be a filtration.

Show that $\mathcal{F}_{T}$ defined in (2.6) is a $\sigma$-algebra.

Prove all parts of Proposition 3.8.

Proposition 3.8: Suppose $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions. Then

1. $T$ is a stopping time if and only if $(T \leq t) \in \mathcal{F}_{t}$ for all $t$.

2. If $T=t$, a.s., then $T$ is a stopping time.

3. If $S$ and $T$ are stopping times, then so are $S \vee T$ and $S \wedge T$.

4. If $T_{n}, n=1,2, \ldots$, are stopping times with $T_{1} \leq T_{2} \leq \cdots$, then so is $\sup_{n} T_{n}$.

5. If $T_{n}, n=1,2, \ldots$, are stopping times with $T_{1} \geq T_{2} \geq \cdots$, then so is $\inf_{n} T_{n}$.

6. If $s \geq 0$ and $S$ is a stopping time, then so is $S+s$.

If $T_{n}$ is defined by (3.2), show $T_{n}$ is a stopping time for each $n$ and $T_{n} \downarrow T$.

This exercise gives an alternate definition of $\mathcal{F}_{T}$ which is more appealing, but not as useful. Suppose that $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions. Show that $\mathcal{F}_{T}$ is equal to the $\sigma$-field generated by the collection of random variables $Y_{T}$ such that $Y$ is a bounded process with paths that are right continuous with left limits and $Y$ is adapted to the filtration $\left\{ \mathcal{F}_{t}\right\}$.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions. Show that if $T$ is a stopping time, then $T$ is $\mathcal{F}_{T}$ measurable.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions and $T$ is a stopping time. Show that if $S$ is a $\mathcal{F}_{T}$ measurable random variable with $S \geq T$, then $S$ is a stopping time.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions. Show that if $S$ and $T$ are stopping times and $X$ is a bounded $\mathcal{F}_{\infty }$ measurable random variable, then

Hint: Let $Y_{t}=\mathbb {E}\left[X \mid \mathcal{F}_{t}\right]$ and $Z_{t}=Y_{t \wedge S}$. Show the left-hand side is equal to $Y_{S \wedge T}$.

Let $W_{t}$ be a Brownian motion and $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration generated by $W$. Let

Show that $T$ is not a stopping time with respect to $\left\{ \mathcal{F}_{t}\right\}$.

Let $X$ be a bounded adapted right-continuous process, let $\varepsilon >0$, let $U_{0}=0$, a.s., and define $U_{i}$ by (16.3) for $i \geq 1$. Show each $U_{i}$ is a stopping time.

Let $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ be a filtration and $\tau$ an optional time with respect to it. Introduce the $\sigma$-algebra

Can one assert that in relation (51) the $\sigma$-algebra $\mathscr {G} = \mathscr {F}_{\sigma +}$?

Relation (51) arises in the proof of Lemma 7. Lemma 7: let $X = \left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a martingale with a.s. right-continuous sample paths, and let $\sigma$ and $\tau$ be optional times (with respect to the filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$) such that $\sigma \leqslant \tau \leqslant c$ a.s., where $c$ is a positive constant; then $\mathbb {E}\left[X_{\tau }\right] = \mathbb {E}\left[X_{\sigma }\right]$. In the proof one sets $\sigma (n) = 2^{-n}\left[2^{n} \sigma +1\right]$, $n \in \mathbb {N}$, where $[\cdot ]$ denotes the integer part of a number, and

where $\overline{\mathscr {A}} = \sigma \left\{ \mathscr {A}, \mathscr {N}\right\}$ for a $\sigma$-algebra $\mathscr {A}$, and $\mathscr {N}$ is the class of $\mathbb {P}$-null sets.

This is a very simple exercise designed to clarify confusion about the roles of past, present, and future in stopping rules. Let $\left\{ X_{n}; n \geq 1\right\}$ be a sequence of IID binary rv's, each with the pmf $\mathsf{p}_{X}(1) = 1/2$, $\mathsf{p}_{X}(0) = 1/2$. Let $J$ be a positive integer-valued rv that takes on the sample value $n$ of the first trial for which $X_{n} = 1$. That is, for each $n \geq 1$,

Let $\tau$ be a stopping time for the natural filtration generated by a sequence of random variables $\left\{ Z_{n}: n \in \mathbb {N}\right\}$. Show that $\mathcal{F}_{\tau } = \sigma \left\{ Z_{\tau \wedge n}: n \in \mathbb {N}\right\}$.

Let $\left\{ X_{t}: t \in \mathbb {R}^{+}\right\}$ be a process with continuous sample paths, adapted to a filtration $\left\{ \mathcal{F}_{t}: t \in \mathbb {R}^{+}\right\}$. Define $\tau (\omega ): = \inf \left\{ t \in \mathbb {R}^{+}: X_{t}(\omega ) \in F\right\}$, for some closed set $F$. Show that $\left\{ \tau \leq t\right\} = \left\{ \omega : \inf_{s \in Q_{t}} d\left(X_{s}(\omega ), F\right) = 0\right\}$, where $Q_{t}$ denotes the set of rational numbers in $[0, t]$. Deduce that $\tau$ is a stopping time for the filtration.

Let $\left\{ \mathscr {F}_{n}\right\}$ be a nondecreasing sequence of $\sigma$-algebras, $\left\{ X_{n}\right\}$ a sequence of random variables such that $X_{n}$ is measurable with respect to $\mathscr {F}_{n}$. Let $B$ be an arbitrary Borel set on the line. Prove that the first hitting time of the set $B$: $\tau_{B} = \inf \left\{ n \geqslant 0: X_{n} \in B\right\}$ is a Markov time.

Let $\tau$ and $\sigma$ be Markov times. Prove that $\tau +\sigma$, $\min \left\{ \tau , \sigma \right\} , \max \left\{ \tau , \sigma \right\}$ are also Markov times with respect to the same sequence of $\sigma$-algebras as $\tau , \sigma$.

Let $\tau$ and $\sigma$ be Markov times. Is the random variable $\tau -\sigma$ a Markov time?

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $\sigma , \tau$ be two stopping times such that $\mathbb {E} \tau , \mathbb {E} \sigma < \infty$.