Math Problems Vault

Problem 7.1

Here you are asked to find a more precise description of the modulus of continuity of Brownian paths. Prove that

\lim _{\delta \rightarrow 0} \sup _{s, t \in [0,1], 0<|t-s|<\delta } \frac{\left|W_{t}-W_{s}\right|}{\sqrt{\delta \log (1 / \delta )}}<\infty , \quad \text{ a.s. }

Hint: Imitate the proof of Theorem 7.1.

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Problem 7.2

The following is part of what is known as Chung's law of the iterated logarithm. We will see in Section 40.3 that there exists $c_{1}$ such that

\mathbb {P}\left(\sup _{s \leq t}\left|W_{s}\right| \leq \lambda \right) \leq c_{1} e^{-\pi ^{2} t / 8 \lambda ^{2}}

for $t / \lambda^{2}$ sufficiently large. Prove that

\liminf _{t \rightarrow \infty } \frac{\sup _{s \leq t}\left|W_{s}\right|}{\sqrt{t / \log \log t}}<\infty , \quad \text{ a.s. }

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Problem 7.3

Let $W_{t}$ be a one-dimensional Brownian motion. We will see in Section 40.3 that there exists $c_{2}$ such that

\mathbb {P}\left(\sup _{s \leq t}\left|W_{s}\right| \leq \lambda \right) \geq c_{2} e^{-\pi ^{2} t / \beta \lambda ^{2}}

if $t / \lambda^{2}$ is sufficiently large. Prove that

\liminf _{t \rightarrow \infty } \frac{\sup _{s \leq t}\left|W_{s}\right|}{\sqrt{t / \log \log t}}>0, \quad \text{ a.s. }

This is the other half of Chung's law of the iterated logarithm. In fact,

\liminf _{t \rightarrow \infty } \frac{\sup _{s \leq t}\left|W_{s}\right|}{\sqrt{t / \log \log t}}=c, \quad \text{ a.s. } \tag {7.6}

Identify $c$ and prove (7.6).

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Problem 7.4

A function $f$ is Hölder continuous of order $\alpha$ at a point $t$ if there exists $c$ such that $|f(u)-f(t)| \leq$ $c|u-t|^{\alpha }$ for all $u$ . Suppose $\alpha >1 / 2$ and $W_{t}$ is a Brownian motion. Show that the event

A=\{ \exists t \in [0,1]: W \text{ is Hölder continuous of order } \alpha \text{ at } t)

has probability 0 .

Hint: Imitate the proof of nowhere differentiability, but use more than three time intervals.

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Problem 7.5

Let $W$ be a one-dimensional Brownian motion and let $M_{t}=\sup_{s \leq t} W_{s}$ (with no absolute value signs). Prove that if $\varepsilon >0$ , then

\liminf _{t \rightarrow \infty } \frac{M_{t}}{\sqrt{t} /(\log t)^{1+\varepsilon }}>0, \quad \text{ a.s. }

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Problem 7.6

This is a complement to Exercise 4.10. Prove that if $p>2$ and $W$ is a Brownian motion, then the $p$ -variation of $W$ , defined in Exercise 4.10 , is finite, a.s.

Hint: Use the fact that the paths of Brownian motion are Hölder continuous of order $\alpha$ if $\alpha <1 / 2$ .

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Problem 7.7

(1)(2)

Let $W$ be a Brownian motion and let $Z$ be the zero set: $Z=\left\{ t \in [0,1]: W_{t}=0\right\}$ .

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(1)

Show there exists a constant $c$ not depending on $x$ or $\delta$ such that

\mathbb {P}\left(\exists s \leq \delta : W_{s}=-x\right) \leq \mathbb {P}\left(\sup _{s \leq \delta }\left|W_{s}\right| \geq |x|\right) \leq c e^{-x^{2} / 2 \delta }

(2)

Use the Markov property of Brownian motion to show that there exists a constant $c$ not depending on $s$ or $t$ such that

\mathbb {P}\left(Z \cap [s, t] \neq \emptyset \right) \leq c\left(1 \wedge \sqrt{\frac{t-s}{s}}\right)

Problem 7.8

(1)(2)

Given a Borel measurable subset $A$ of $[0,1]$ , define

H_{\gamma }(A)=\lim _{\delta \rightarrow 0} \sup \left[\inf \left\{ \sum _{i=1}^{\infty }\left[b_{i}-a_{i}\right]^{\gamma }: A \subset \cup _{i=1}^{\infty }\left[a_{i}, b_{i}\right], \sup _{i}\left|b_{i}-a_{i}\right| \leq \delta \right\} \right]

In other words, cover $A$ by the union of intervals $\left[a_{i}, b_{i}\right]$ and define the analog of Lebesgue measure. The differences are that we look at $\left|b_{i}-a_{i}\right|^{\gamma }$ but do not require that $\gamma$ be one, and we require that none of the intervals be longer than $\delta$ . The quantity $H_{\gamma }(A)$ is called the Hausdorff measure of $A$ with respect to the function $x^{\gamma }$ . The Hausdorff dimension of a set $A$ is defined to be

\inf \left\{ \gamma : H_{\gamma }(A)>0\right\} =\sup \left\{ \gamma : H_{\gamma }(A)=\infty \right\}

(For subsets of $\mathbb {R}^{d}$ , we replace the intervals $\left[a_{i}, b_{i}\right]$ by balls of radius $r_{i \cdot }$ .) As a warm-up to this exercise, prove that the Hausdorff dimension of the standard Cantor set in $[0,1]$ is $\log 2 / \log 3$ .

The purpose of this exercise is to show that if $W$ is a Brownian motion and $Z=\{ t \in [0,1]$ : $\left.W_{t}=0\right\}$ is the zero set, then the Hausdorff dimension of $Z$ is no more than $1 / 2$ .

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(1)

For each $n$ , let $\mathcal{C}_{n}$ be the collection of intervals $\left[i / 2^{n},(i+1) / 2^{n}\right]$ contained in $[0,1]$ that intersect $Z$ . $\left(\mathcal{C}_{n}\right.$ is random.) If $\# \mathcal{C}_{n}$ is the cardinality of $\mathcal{C}_{n}$ , use Exercise 7.7 to show

\mathbb {E}\left[\# \mathcal{C}_{n}\right] \leq \sum _{i=0}^{2^{n}-1} \mathbb {P}\left(Z \cap \left[i / 2^{n},(i+1) / 2^{n}\right] \neq \emptyset \right) \leq c 2^{n / 2}

(2)

Write

\sum _{\left[i / 2^{n},(i+1) / 2^{n}\right] \in \mathcal{C}_{n}}\left|2^{-n}\right|^{\gamma }=2^{-n \gamma } \# \mathcal{C}_{n}

Use the Chebyshev inequality and (1) to conclude that the Hausdorff dimension of $Z$ is less than or equal to $1 / 2$ , a.s. (We will show that it is at least $1 / 2$ in Exercise 14.10.)