Math Problems Vault

Problem 2.1

Suppose $W$ is a Brownian motion on $[0,1]$ . Let

Y_{t}=W_{1-t}-W_{1} .

Show that $Y_{t}$ is a Brownian motion on $[0,1]$ .

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Problem 2.2

This exercise shows that the projection of a $d$ -dimensional Brownian motion onto a hyperplane yields a one-dimensional Brownian motion. Suppose $\left(W_{t}^{(1)}, \ldots , W_{t}^{(d)}\right)$ is a $d$ -dimensional Brownian motion started from 0 and $\lambda_{1}, \ldots , \lambda_{d} \in \mathbb {R}$ with $\sum_{i=1}^{d} \lambda_{i}^{2}=1$ . Show that $X_{t}=\sum_{i=1}^{d} \lambda_{i} W_{t}^{(i)}$ is a one-dimensional Brownian motion started from 0.

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Problem 2.3

This exercise shows that rotating a Brownian motion about the origin yields another Brownian motion. Let $W$ be a $d$ -dimensional Brownian motion started at 0 and let $A$ be a $d \times d$ orthogonal matrix, that is, $A^{-1}=A^{\mathrm{T}}$ . Show that $Y_{t}=A W_{t}$ is again a $d$ -dimensional Brownian motion.

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Problem 2.4

(1)(2)(3)

Here is a converse to Exercise 2.2: roughly speaking, if all the projections of a $d$ -dimensional process $X$ onto hyperplanes are one-dimensional Brownian motions, then $X$ is a $d$ -dimensional Brownian motion.

Suppose $\left(X_{t}^{1}, \ldots , X_{t}^{d}\right)$ is a $d$ -dimensional continuous process, i.e., one taking values in $\mathbb {R}^{d}$ . Let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration generated by $X$ . Suppose that whenever $\lambda_{1}, \ldots , \lambda_{d} \in \mathbb {R}$ with $\sum_{i=1}^{d} \lambda_{i}^{2}=1$ , then $\sum_{i=1}^{d} \lambda_{i} X_{t}^{i}$ is a one-dimensional Brownian motion started at 0 with respect to the filtration $\left\{ \mathcal{F}_{t}\right\}$ .

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(1)

If $u=\left(u_{1}, \ldots , u_{d}\right)$ , let $\| u\| =\left(\sum u_{j}^{2}\right)^{1 / 2}$ and let $\lambda_{j}=u_{j} /\| u\|$ . Calculate

\mathbb {E}\left[\mathrm{exp}\left(i \sum _{j=1}^{d} u_{j} X_{t}^{j}\right)\right]=\mathbb {E}\left[\mathrm{exp}\left(i\| u\| \sum _{j=1}^{d} \lambda _{j} X_{t}^{j}\right)\right]

the joint characteristic function of $X_{t}$ .

(2)

If $t_{0}<t_{1}<\cdots <t_{n}$ , use independence and (1) to calculate

\mathbb {E}\left[\mathrm{exp}\left(i \sum _{k=0}^{n-1} \sum _{j=1}^{d} u_{j}^{k}\left(X_{t_{k+1}}^{j}-X_{t_{k}}^{j}\right)\right)\right]

(3)

Prove that $\left(X_{t}^{1}, \ldots , X_{t}^{d}\right)$ is a $d$ -dimensional Brownian motion started from 0 .

(Some care is needed with the filtrations. If we only know that $Y^{\lambda }=\sum_{i} \lambda_{i} X^{i}$ is a Brownian motion with respect to the filtration generated by $Y^{\lambda }$ for each $\lambda =\left(\lambda_{1}, \ldots , \lambda_{d}\right)$ , the assertion is not true. See Revuz and Yor (1999), Exercise I.1.19.)

Problem 2.5

Let $W_{t}$ be a Brownian motion and suppose

\lim _{t \rightarrow \infty } W_{t} / t=0 \text{, } \quad \text{ a.s. } \tag {2.5}

Let $Z_{t}=t W_{1 / t}$ if $t>0$ and set $Z_{0}=0$ . (This is called time inversion.) Show that $Z$ is a Brownian motion. (We will see later that the assumption (2.5) is superfluous; see Theorem 7.2.)

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Problem 2.6

Let $X$ and $Y$ be two independent Brownian motions started at 0 and let $t_{0}>0$ . Let

Z_{t}= \begin{cases} X_{t}, & t \leq t_{0}, \\ X_{t_{0}}+Y_{t-t_{0}}, & t>t_{0} .\end{cases}

Prove that $Z$ is also a Brownian motion.

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Problem 2.7

Let $\mathcal{F}_{t}^{00}$ and $\mathcal{F}_{t}^{0}$ be defined as in (1.1) and (1.2). Prove that if $X$ is $\mathcal{F}_{t}^{0}$ measurable, there exists $Z$ such that $Z$ is $\mathcal{F}_{t}^{00}$ measurable and $Y=Z$ , a.s.

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Problem 2.8

Let $\mathcal{F}_{t}^{00}$ and $\mathcal{F}_{t}^{0}$ be defined as in (1.1) and (1.2). The symmetric difference of two sets $A$ and $B$ is defined by $A \bigtriangleup B=(A \backslash B) \cup (B \backslash A)$ . Prove that

\mathcal{F}_{t}^{0}=\left\{ A \subset \Omega : A \bigtriangleup B \in \mathcal{N} \text{ for some } B \in \mathcal{F}_{t}^{00}\right\} .

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