Verifying That a Given Time Is a Stopping Time

Construct an example of a real-valued process $X = \left\{ X(t), t \geqslant 0\right\}$ of càdlàg type (but not continuous) and a closed set $F \subset \mathbb {R}$ such that $\tau_{\infty } < \tau_{F}$ a.s., where $\tau_{\infty }$ and $\tau_{F}$ are defined in the proof of Theorem 3.

Theorem 3. Let $X = \left\{ X(t), t \geqslant 0\right\}$ be a random function taking values (for each $t \geqslant 0$) in a metric space $(\mathrm{S}, \rho )$. Suppose the sample paths of $X$ are almost surely continuous. Then for every closed set $F \subset \mathrm{S}$ the time

is a Markov time with respect to the natural filtration $\mathbb {F}^{X}$ of the process $X$ (it is assumed that $\tau_{F}(\omega ) = \infty$ if $X(t, \omega ) \notin F$ for all $t \geqslant 0$).

In the proof of Theorem 3 one sets $\tau_{n} := \tau_{G_{n}} = \inf \left\{ t \geqslant 0: X(t, \omega ) \in G_{n}\right\}$, where the open sets $G_{n} = \left\{ x \in \mathrm{S}: \rho (x, F) < 1 / n\right\}$ form, as $n$ grows, shrinking neighborhoods of $F$. Clearly, $F \subset G_{n+1} \subset G_{n}$, whence $\tau_{n}(\omega ) \leqslant \tau_{n+1}(\omega ) \leqslant \tau_{F}(\omega )$ for all $n \in \mathbb {N}$, $\omega \in \Omega$. Therefore for each $\omega$ there exists

Let $X = \left\{ X_{n}, n \in \mathbb {N}\right\}$ be a sequence of real-valued random variables and let a set $B \in \mathscr {B}(\mathbb {R})$ be given. Let $\tau = \tau (\omega )$ be the first hitting time of the set $B$, defined by formula (I.49).

Formula (I.49):

($\tau (\omega ) := \infty$ if $X_{n}(\omega ) \notin B$ for all $n \in \mathbb {N}$).

We have already seen (see §3) that $\tau$ is a Markov time with respect to the natural filtration of the process $X$. Now introduce

(in the definition of the times $\tau$ and $\sigma$ it is assumed that if the set in curly braces is empty for some point $\omega$, then the value of the quantity under consideration at this point is taken to be $+\infty$ ). Prove that $\sigma$ is a Markov time with respect to the natural filtration of the process $X$. Construct an example of a sequence $X$ and a set $B$ for which the time $\gamma$ is a Markov time with respect to the natural filtration, and also an example of $X$ and $B$ for which $\gamma$ is not such a time.

Let $X_{1}, X_{2}, \ldots$ be a sequence of non-negative independent random variables and let $N(t) = \max \left\{ n: X_{1}+X_{2}+\cdots +X_{n} \leq t\right\}$. Show that $N(t)+1$ is a stopping time with respect to a suitable filtration to be specified.

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple symmetric random walk on the positive and negative integers, with $S_{0} = 0$. For each of the following random variables, determine whether or not it is a stopping time and find its mean:

Let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $d$ - dimensional stochastic process and $A, A_{n}, C \in \mathscr {B}\left(\mathbb {R}^{d}\right), n \geqslant 1$. Then

Let $U \subset \mathbb {R}^{d}$ be an open set and assume that $\left(X_{t}\right)_{t \geqslant 0}$ is a stochastic process with continuous paths. Show that $\tau_{U} = \tau_{U}^{\circ }$.

Let $\left(I_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a continuous adapted process with values in $[0, \infty )$ and a.s. increasing sample paths. Set for $u \geqslant 0$

Show that

Show that, in Theorem 19.17, the quadratic variation $\left\langle M\right\rangle_{t}$ is a $\mathscr {G}_{t}$ stopping time.