Math Problems Vault

Problem 4.1

If $W$ is a Brownian motion, let $S_{t}=\sup_{s \leq t} W_{s}$ . Find the density for $S_{t}$ .

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Problem 4.2

With $W$ and $S$ as in Exercise 4.1, find the joint density of $\left(S_{t}, W_{t}\right)$ .

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Problem 4.3

Let $W$ be a Brownian motion started at $a>0$ and let $T_{0}$ be the first time $W$ hits $0$ . Find the law of $\sup_{t \leq T_{0}} W_{t}$ .

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Problem 4.4

Use the reflection principle to prove that if $W$ is a Brownian motion and $T=\inf \left\{ t>0: W_{t} \in \right.$ $(0, \infty )\}$ , then

\mathbb {P}\left(T=0\right)=1

In other words, Brownian motion enters the interval $(0, \infty )$ immediately. By symmetry it enters the interval $(-\infty , 0)$ immediately. Conclude that Brownian motion hits 0 infinitely often in every time interval $[0, t]$ .

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Problem 4.5

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

T=\inf \left\{ t>0: W_{t}=\sup _{0 \leq s \leq 1} W_{s}\right\}

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

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Problem 4.6

(1)(2)

Let $W$ and $S$ be as in Exercise 4.1.

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(1)

Let $0<s<t<u$ and let $a<b$ with $b-a \leq 1$ . Show that there exists a constant $c$ , depending on $s, t$ , and $u$ , but not $a$ or $b$ , such that

\mathbb {P}\left(S_{s} \in [a, b], \sup _{t \leq r \leq u} W_{r} \in [a, b]\right) \leq c(b-a)^{2}

(2)

Show that the path of a Brownian motion does not take on the same value as a local maximum twice. That is, if $S$ and $T$ are times when $W$ has a local maximum, then $W_{S} \neq W_{T}$ , a.s.

Problem 4.7

Let $V_{t}$ be the number of upcrossings of $[0,1]$ by a Brownian motion $W$ up to time $t$ . This means we let $S_{1}=0, T_{i}=\inf \left\{ t>S_{i}: W_{t} \geq 1\right\}$ , and $S_{i+1}=\inf \left\{ T>T_{i}: W_{t} \leq 0\right\}$ for $i=1,2, \ldots$ , and we set $V_{t}=\sup \left\{ k: T_{k} \leq t\right\}$ . Show that $V_{t} \rightarrow \infty$ , a.s., as $t \rightarrow \infty$ .

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Problem 4.8

(1)(2)

Let $W$ be a Brownian motion. The zero set of Brownian motion is the random set

Z(\omega )=\left\{ t \in [0,1]: W_{t}(\omega )=0\right\}

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(1)

Show that $Z(\omega )$ is a closed set for each $\omega$ .

(2)

Show that with probability one, every point of $Z(\omega )$ is a limit point of $Z(\omega )$ . Conclude that $Z(\omega )$ is an uncountable set.

Problem 4.9

Let $W$ be a one-dimensional Brownian motion and $\delta >0$ .

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(1)

Prove that there exists $\gamma$ such that if $t \leq \gamma$ , then

\mathbb {P}\left(0 \leq W_{t} \leq \delta / 2\right) \geq 1 / 4 \quad \text{ and } \quad \mathbb {P}\left(-\delta / 2 \leq W_{t} \leq 0\right) \geq 1 / 4

(2)

Prove there exists $\gamma$ such that

\mathbb {P}\left(\sup _{s \leq \gamma }\left|W_{s}\right|>\delta / 2\right) \leq 1 / 8

(3)

Prove that if $m \geq 1$ , then

\begin{aligned} & \mathbb {P}\left( \sup _{m \gamma \leq s \leq (m+1) \gamma }\left|W_{s}-W_{m \gamma }\right| \leq \delta / 2, W_{m \gamma } \in [0, \delta / 2],\left|W_{(m+1) \gamma }\right| \leq \delta / 2 \mid \mathcal{F}_{m \gamma } \right) \\ & \geq \frac{1}{8} \mathbb {P}\left(\sup _{m \gamma \leq s \leq (m+1) \gamma }\left|W_{s}-W_{m \gamma }\right| \leq \delta / 2, W_{m \gamma } \in [0, \delta / 2]\right) \end{aligned}

and the same with $W_{m \gamma } \in [-\delta / 2,0]$ in place of $W_{m \gamma } \in [0, \delta / 2]$ . Conclude that

\begin{aligned} & \mathbb {P}\left( \sup _{m \gamma \leq s \leq (m+1) \gamma }\left|W_{s}-W_{m \gamma }\right| \leq \delta / 2,\left|W_{m \gamma }\right| \leq \delta / 2,\left|W_{(m+1) \gamma }\right| \leq \delta / 2 \mid \mathcal{F}_{m \gamma } \right) \\ & \geq \frac{1}{8} \mathbb {P}\left(\sup _{m \gamma \leq s \leq (m+1) \gamma }\left|W_{s}-W_{m \gamma }\right| \leq \delta / 2,\left|W_{m \gamma }\right| \leq \delta / 2\right) \end{aligned}

(4)

Use induction to prove that if $t_{0}>0$ , there exists $c_{1}>0$ such that

\mathbb {P}\left(\sup _{s \leq t_{0}}\left|W_{s}\right| \leq \delta \right)>c_{1}

(5)

Prove that if $W$ is a $d$ -dimensional Brownian motion, $t_{0}>0$ , and $\delta >0$ , there exists $c_{2}$ such that

\mathbb {P}\left(\sup _{s \leq t_{0}}\left|W_{s}\right| \leq \delta \right)>c_{2} .

Problem 4.10

The $p$ -variation of a function $f$ on the interval $[0,1]$ is defined by

V^{p}(f)=\sup \left\{ \sum _{i=0}^{n-1}\left|f\left(t_{i+1}\right)-f\left(t_{i}\right)\right|^{p}: n \geq 1,0=t_{0}<t_{1}, \cdots <t_{n}=1\right\}

the supremum is over all partitions $P$ of $[0,1]$ . In this exercise we will prove that if $p<2$ and $W$ is a Brownian motion, then $V^{p}(W)=\infty$ , a.s.

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(1)

Let $X_{i}$ be an i.i.d. sequence of random variables with finite mean. Use the strong law of large numbers to prove that if $K>\mathbb {E}\left[X_{1}\right]$ , then

\mathbb {P}\left(\sum _{i=1}^{n} X_{i}>K n\right) \rightarrow 0

as $n \rightarrow \infty$ .

(2)

If $p<2$ , take $r \in (p, 2)$ , and let $\varepsilon_{n}=n^{-1 / r}$ . Let $S_{0}=0$ and for $i \geq 0$ , set $S_{i+1}=\inf \left\{ t>S_{i}:\left|W_{t}-W_{S_{i}}\right|>\varepsilon_{n}\right\}$ . Set $X_{i}=\varepsilon_{n}^{-2}\left(S_{i}-S_{i-1}\right)$ . Prove that the $X_{i}$ are i.i.d. with finite mean.

(3)

Use (1) to show that

\mathbb {P}\left(S_{n}>1\right)=\mathbb {P}\left(\sum _{i=1}^{n} X_{i}>\varepsilon _{n}^{-2}\right) \rightarrow 0

as $n \rightarrow \infty$ .

(4)

Using the partition $\left\{ S_{0}, S_{1}, \ldots , S_{n}\right\}$ , show that $V^{p}(W) \geq n \varepsilon_{n}^{p}$ on the event $\left(S_{n} \leq 1\right)$ .

(5)

Conclude $V^{p}(W)=\infty$ , a.s.