Sample path properties of Brownian motion

Let $B(\cdot )$ be SBM. Let $M_{n} = \sup \left\{ \left|B(t)-B(n)\right|: n-1 \leq t \leq n\right\}$, $n = 1,2, \ldots$

Let $W_{t}$ be a Brownian motion. Show that if $\alpha >1 / 2$, then

Hint: Let $\alpha_{0} \in (1 / 2, \alpha )$, estimate

using (3.9), and then use the Borel-Cantelli lemma.

Let $W_{t}$ be a one-dimensional Brownian motion and $\alpha \in (0,1 / 2]$. Prove that

If $W$ is a Brownian motion and $b$ is a constant, then the process $X_{t}=W_{t}+b t$ is a Brownian motion with drift. Prove that if $b>0$, then

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

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 $[0, t]$.

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

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 $t \rightarrow \infty$.

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

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

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

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.

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

Hint: Imitate the proof of Theorem 7.1.

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

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

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

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

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

Identify $c$ and prove (7.6).

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

has probability 0 .

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

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

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 $\alpha <1 / 2$.

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

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

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

(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 $1 / 2$.

Suppose $W_{t}$ is a Brownian motion and $a \in \mathbb {R}$. Show that the amount of time Brownian motion spends at the point $a$ is zero, i.e., that

for all $t>0$.

Use the Blumenthal $0-1$ law to show that if $W$ is a one-dimensional Brownian motion and $T=\inf \left\{ t>0: W_{t}>0\right\}$ is the first time Brownian motion hits $(0, \infty )$, then $\mathbb {P}\left(T=0\right)=1$.

Show by a direct argument that $W(\cdot , \omega )$ is with probability $\mathbf{1}$ of unbounded variation on $[0,1]$ : Let $Y_{n}=\sum_{i=1}^{2^{n}}\left|W\left(i 2^{-n}\right)-W\left((i-1) 2^{-n}\right)\right|$. Show that $Y_{n}$ has mean $2^{n / 2} \mathbb {E}\left[\left|W_{1}\right|\right]$ and variance at most $\operatorname {Var}\left[\left|W_{1}\right|\right]$. Conclude that $\sum \mathbb {P}\left(Y_{n}<n\right)<\infty$.

Extend Theorem 37.4 to the set $[t: W(t, \omega )=\alpha ]$.

Show by means of the transformation (37.12) that for positive $a$ and $b$ the probability is 1 that the process is within the boundary $-a t<W_{1}<b t$ for all sufficiently large $t$. Show that $a /(a+b)$ is the probability that it last touches above rather than below.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Let $\left\{ t_{n}\right\}_{n \geqslant 1}$ be a sequence of positive numbers such that $\sum_{n = 1}^{\infty } t_{n}^{-1 / 2} < \infty$. Prove that then $\left|W\left(t_{n}\right)\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Prove that almost all sample paths of the Wiener process are Hölder functions with exponent $\gamma < 1 / 2$, i.e. on every time interval $[a, b] \subset [0, \infty )$

(A proof of this statement can be found, e.g., in [12], Theorem 3 on p. 127.) Can the constant $C_{\gamma }$ in the above result be taken independent of the interval $[a, b]$ ?

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Prove that with probability one

(thus the Hölder condition of order $1 / 2$ fails at the point 0 ).

Using the reasoning carried out in the proof of Theorem 1, prove that the Wiener process $W = \left\{ W(t), 0 \leqslant t \leqslant 1\right\}$ has sample paths that a.s. do not satisfy at any point the Hölder condition with exponent $\gamma > 1 / 2$.

Theorem 1 (Paley-Wiener-Zygmund). With probability one the sample paths of Brownian motion $W = \left\{ W(t), t \geqslant 0\right\}$ are not differentiable at any point $t$ of the half-line $[0, \infty )$.

Derive the result of Exercise 5 from Theorem 19 of Chapter II and the property that for $q \in \mathbb {N}, 0 \leqslant s < t < \infty$

Recall that for a set $U \subset T(U$ may coincide with $T)$, where $(T, \delta )$ is a pseudometric space, an $\varepsilon$-net is a set $S_{\varepsilon } \subset T$ such that for every $t \in U$ there exists an element $s \in S_{\varepsilon }$ for which $\delta (s, t) \leqslant \varepsilon$. In other words, $U$ is covered by the closed balls

If $U$ has a finite $\varepsilon$-net, then we call a minimal $\varepsilon$-net any of its $\varepsilon$-nets $S_{\varepsilon }^{\min }(U)$ such that $\left(S_{\varepsilon }^{\min }(U)\right)$ is minimal, where $(V)$ denotes the number of elements of a finite set $V$. Put $N_{\delta }(\varepsilon , U) = \# \left(S_{\varepsilon }^{\min }(U)\right)$, emphasizing the dependence on the pseudometric $\delta$ as well.

Following [153], we introduce two conditions on the space ( $T, \delta$ ). Suppose there exist $d \in \mathbb {N}$ and $a > 0$ such that for each $U \subset T$ with diameter $D_{U}: = \sup \left\{ \delta (s, t): s, t \in U\right\} < \infty$ and any $\varepsilon > 0$

Suppose there exists $b > 0$ such that for any minimal $\varepsilon$-net $S_{\varepsilon }^{\min }(T)$

where $B_{5 \varepsilon }(t)$ is defined according to (II.53).

Theorem 19 of Chapter II (see [153; p. 134]). Let $X = \left\{ X(t), t \in T\right\}$ be a random function with values in a Polish space ( $\mathrm{S}, \rho$ ) for each $t \in T$, defined on some $(\Omega , \mathscr {F}, \mathbb {P})$ and a pseudometric space $(T, \delta )$ satisfying conditions (II.55) and (II.56). Suppose that for some $\gamma \in (0,1)$, $\alpha > d / \gamma$ and $\beta > 0$

Then $X$ has a modification $Y = \left\{ Y(t), t \in T\right\}$ with continuous sample paths such that for any $\lambda \in (0, \gamma -d / \alpha )$

and for each $t_{0} \in T$

Let $W$ be a standard Wiener process and let $F(u, v)$ be the event that $W$ has no zero in the interval $(u, v)$.

Let $W$ be a standard Wiener process. Show that

Set $t = 2^{n}$ and $w = 2^{2 n / 3}$ and use the Borel-Cantelli lemma to show that $t^{-1} W(t) \rightarrow 0$ a.s. as $t \rightarrow \infty$.

Let $W$ be a standard Wiener process, $u \in \mathbb {R}$, and let $Z = \left\{ t: W(t) = u\right\}$. Show that $Z$ is a null set (i.e. has Lebesgue measure zero) with probability one.

Let $M(t) = \max_{0 \leq s \leq t} W(s)$, where $W$ is a standard Wiener process. Show that $M(t)$ is attained at exactly one point in $[0, t]$, with probability one.

Show that the total variation of a standard Wiener process over any non-trivial interval does not exist. [The total variation $V_{a, b}(f)$ of a function $f: \mathbb {R} \rightarrow \mathbb {R}$ over the bounded interval $[a, b]$ is defined as

where the supremum is taken over all increasing sequences $s_{0} = a, s_{1}, s_{2}, \ldots , s_{n} = b$. You may find it useful that the quadratic variation of the Wiener process exists and is non-zero almost surely.]

Prove that the Wiener process is not differentiable in probability.

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Prove that with probability 1 its trajectory has unbounded variation on an arbitrary interval $[a, b] \subset \mathbb {R}_{+}$, i.e., that

where $T=\left\{ a=t_{0}<\ldots <t_{n}=b\right\}$ is a partition of the interval $[a, b]$.

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process. For each $n \in \mathbb {N}$ consider partitions of the interval $[a ; b]$ with points

and

Set

Find the mean-square (i.e., $L^2$) limit of the quantities $U_{n}$ as $n \rightarrow +\infty$.

Let the sequence of positive numbers $\left\{ t_{n}, n \in \mathbb {N}\right\}$ be such that $\sum_{n=1}^{\infty } t_{n}^{-1 / 2}<\infty$. Prove that then $\left|W_{t_{n}}\right| \rightarrow +\infty$ a.s. as $n$ increases. Here $\left(W_{t}, t \geq 0\right)$ is a Wiener process.

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process. Prove that for any $0 \leq a<b \leq c<d$, with probability 1 we have

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process, and let $s < u < 0$. Find

Let $\left\{ \left(B_{t}, \mathcal{F}_{t}\right): 0 \leq t \leq 1\right\}$ be a Brownian motion with continuous sample paths. Define $\tau (\omega )$ as the smallest $t$ at which the sample path $B(\cdot , \omega )$ achieves its maximum value, $M(\omega )$. (Note: $\tau$ is not a stopping time.) Follow these steps to show that the sets $\Omega_{t}: = \left\{ \tau = t\right\}$ for $0 \leq t \leq 1$ is a family suitable for the construction in Example $\langle 2 \rangle$.

Example 2: given any uncountable partition $\left\{ \Omega_{t}: 0 \leq t \leq 1\right\}$ of $\Omega$ into $\mathcal{F}_{1}$-measurable sets with $\mathbb {P}\left(\Omega_{t}\right) = 0$ for every $t$ (``a family suitable for the construction''), one can build a badly-behaved modification $B^{*}$ of $B$, equal to $B$ almost surely at each fixed $t$ and satisfying the same finite-dimensional distributions, but with $\sup_{t}\left|B_{t}^{*}(\omega )\right|$ nonmeasurable, by reassigning $B^{*}(t, \omega )$ to an arbitrarily large nonmeasurable value on each $\Omega_{t}$.

Prove that a Wiener process is not differentiable in probability.

Prove that almost all trajectories of a Wiener process are nowhere differentiable.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}, f: \mathbb {R}^{d} \rightarrow \mathbb {R}$ be a continuous function such that $\int_{0}^{t} f\left(B_{s}\right) d s = 0$ for all $t > 0$. Show that $f\left(B_{s}\right) = 0$ for all $s > 0$, and conclude that $f \equiv 0$.

Use the LIL (Corollary 12.2) to give an alternative proof for the fact that a one - dimensional Brownian motion oscillates in every time interval $[0, \epsilon ]$ infinitely often around its starting point.

Show that $\mathrm{BM}^{1}$ is almost surely not $1 / 2$ - Hölder continuous:

The proof of Theorem 10.3 actually shows, that almost all Brownian paths are nowhere Lipschitz continuous. Modify the argument of this proof to show that almost all Brownian paths are nowhere Hölder continuous for all $\alpha > 1 / 2$. Why does the argument break down for $\alpha = 1 / 2$ ?

Use Theorem 10.6 to show that the strong $p$ - variation $\operatorname {VAR}_{p}(B ;[0,1])$ of $\mathrm{BM}^{1}$ is for $p > 2$ finite.