Math Problems Vault

Problem 6.1

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Let $Z_{0}, Z_{1}, Z_{2}, \ldots$ be a sequence of independent identically distributed mean zero normal random variables with variance one. Define

X_{t}=\frac{t^{2}}{2 \sqrt{\pi }} Z_{0}+\sum _{k=1}^{\infty }\left(\sqrt{\frac{2}{\pi }} \frac{\cos k t}{k^{2}}\right) Z_{k} \tag {6.9}

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(1)

Show that the convergence in (6.9) is absolute and uniform over $t \in [0,1]$ .

(2)

Show that $X_{t}$ is a Gaussian process.

(3)

If $W_{t}$ is a Brownian motion and

Y_{t}=\int _{0}^{t} W_{r} d r, \quad t \in [0,1]

show that $X$ and $Y$ have the same finite-dimensional distributions. Show that $X$ and $Y$ have the same law when viewed as random variables taking values in $C[0,1]$ . (The process $X$ is sometimes known as integrated Brownian motion.)

(4)

Find $\operatorname {Cov}\left[ X_{s}, X_{t} \right]$ .

Problem 6.2

Let $\left\{ \varphi_{n}\right\}$ be a complete orthonormal system for $L^{2}[0,1]$ . Show that the sum (6.8) converges in $L^{2}$ and give the details of the proof that the resulting process $W$ is a mean zero Gaussian process with $\operatorname {Cov}\left[ W_{s}, W_{t} \right]=s \wedge t$ if $s, t \in [0,1]$ .

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Problem 6.3

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Let $\mathcal{D}=\left\{ k / 2^{n}: n \geq 1, k=0,1, \ldots , 2^{n}\right\}$ be the dyadic rationals. Suppose the collection of random variables $\left\{ V_{t}: t \in \mathcal{D}\right\}$ is jointly normal, each $V_{t}$ has mean zero, and $\operatorname {Cov}\left[ V_{s}, V_{t} \right]=s \wedge t$ .

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(1)

Prove that the paths of $V$ are uniformly continuous over $t \in \mathcal{D}$ .

(2)

If we define $W_{t}=\lim_{s \in \mathcal{D}, s \rightarrow t} V_{s}$ , prove that $W$ is a Brownian motion.

Problem 6.4

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In this and the next exercise we give the Haar function construction of Brownian motion. Let $\varphi_{00}=1$ on $[0,1]$ and for $i=1,2, \ldots$ , and $1 \leq j \leq 2^{i-1}$ , set

\varphi _{i j}(x)= \begin{cases} 2^{(i-1) / 2}, & (2 j-2) / 2^{i} \leq x<(2 j-1) / 2^{i} \\ -2^{(i-1) / 2}, & (2 j-1) / 2^{i} \leq x<2 j / 2^{i} \\ 0, & \text{ otherwise }\end{cases}

It is a well-known and easily proved result from analysis (see, e.g., Bass (1995), Section I.2) that the collection $\left\{ \varphi_{i j}\right\}$ is a complete orthonormal system for $L^{2}[0,1]$ .

For each $i, j$ , define

\psi _{i j}(t)=\int _{0}^{t} \varphi _{i j}(s) d s

for each $i$ and $j$ , let $Y_{i j}$ be independent mean zero normal random variables with variance one, and let

V_{i}(t)=\sum _{j=1}^{2^{i-1}} Y_{i j} \varphi _{i j}(t)

for $i \geq 1$ . Set $V_{0}=Y_{00} \varphi_{00}$ .

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(1)

Fix $i \geq 1$ . Prove that each $\psi_{i j}$ is bounded by $2^{(-i-1) / 2}$ . Prove that the sets $\left\{ t: \psi_{i j}(t)>0\right\}$ , $j=1, \ldots , 2^{i-1}$ , are disjoint.

(2)

Fix $i \geq 1$ . Write

\mathbb {P}\left(\exists t \in [0,1]:\left|V_{i}(t)\right|>i^{-2}\right) \leq \mathbb {P}\left(\exists j \leq 2^{i-1}:\left|Y_{i j}\right| 2^{(-i-1) / 2}>i^{-2}\right),

use Proposition A. 52 to estimate this, and conclude that

\sum _{i=1}^{\infty } \mathbb {P}\left(\sup _{0 \leq t \leq 1}\left|V_{i}(t)\right|>i^{-2}\right)<\infty \tag {6.10}

Problem 6.5

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This is a continuation of Exercise 6.4. With $\varphi_{i j}, \psi_{i j}, Y_{i j}$ , and $V_{i}$ as in that problem, let

W_{t}=\sum _{i=0}^{\infty } V_{i}(t)

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(1)

Prove that $W$ is a jointly normal Gaussian process with mean zero and $\operatorname {Cov}\left[ W_{s}, W_{t} \right]=s \wedge t$ .

(2)

Use (6.10) and the Borel-Cantelli lemma to show that $\sum_{i=1}^{n}\left|V_{i}(t)\right|$ converges uniformly over $[0,1]$ . Conclude that $W$ is a Brownian motion.