Math Problems Vault

Problem 15.1

(1)(2)

Without some supplemental conditions on $T$ , the problem of Skorokhod embedding is trivial. Suppose $W$ is a Brownian motion with respect to a filtration $\left\{ \mathcal{F}_{t}\right\}$ satisfying the usual conditions. Suppose $Y$ is a finite random variable and suppose $h$ is a real-valued function such that $h\left(W_{1}\right)$ has the same law as $Y$ .

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(1)

Show that if $T=\inf \left\{ t>1: W_{t}=h\left(W_{1}\right)\right\}$ , then $W_{T}$ and $Y$ have the same law.

(2)

Give an example of a mean zero random variable $Y$ with finite variance such that if $T$ is defined as in (1), then $\mathbb {E}\left[T\right]=\infty$ .

Problem 15.2

Show that the triple integral on the right-hand side of (15.10) is equal to the expression in $(15.11)$ .

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Problem 15.3

A sketch was given for the proof of Corollary 15.5. Provide a detailed proof.

Corollary 15.5: Let $W$ be a Brownian motion and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration for $W$ . Let $Y$ be a random variable with $\mathbb {E}\left[Y\right]=0$ and $\operatorname {Var}\left[Y\right]<\infty$ . There exists a stopping time $V$ with respect to $\left\{ \mathcal{F}_{t}\right\}$ such that $W_{V}$ has the same law as $Y$ .

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Problem 15.4

(1)(2)(3)

Here is another approach to proving Corollary 15.5. Let $Y, N, T$ , and $\left\{ \mathcal{G}_{t}\right\}$ be as in the proof of Theorem 15.4 .

Theorem 15.4: Suppose $Y$ is a random variable with $\mathbb {E}\left[Y\right]=0$ and $\mathbb {E}\left[Y^{2}\right]<\infty$ . There exists a Brownian motion $N$ and a stopping time $T$ with respect to the minimal augmented filtration $\left\{ \mathcal{G}_{t}\right\}$ of $N$ such that $N_{T}$ is equal in law to $Y$ . Moreover $\mathbb {E}\left[T\right]=\mathbb {E}\left[Y^{2}\right]$ .

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(1)

Show that there is a random variable $U$ that is measurable with respect to $\sigma \left(N_{s}: 0 \leq s<\infty \right)$ such that $U=T$ , a.s.

(2)

Show there is a Borel measurable map $H: C[0, \infty ) \rightarrow [0, \infty )$ such that $U=H(N)$ .

(3)

If $W$ is a Brownian motion, define $V=H(W)$ . Show $V$ is a stopping time with respect to the minimal augmented filtration generated by $W$ such that $W_{V}$ has the same law as $Y$ .

Problem 15.5

Suppose $p \in (0,1 / 2)$ and $Y$ is a random variable such that $\mathbb {P}\left(Y=1\right)=\mathbb {P}\left(Y=-1\right)=p$ and $\mathbb {P}\left(Y=0\right)=1-2 p$ . Let $W$ be a Brownian motion. Let $S_{x}=\inf \left\{ t>0: W_{t}=x\right\}$ and let $T=\inf \left\{ t>S_{x} \wedge S_{-x}: W_{t} \in \left\{ -1,0,1\right\} \right\}$ . Determine $x$ such that $W_{T}$ and $Y$ have the same law.

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Problem 15.6

Suppose $Y$ is a mean zero random variable and there exists a real number $K>0$ such that $\left|Y\right| \leq K$ , a.s. Let $W$ be a Brownian motion and let $T$ be a stopping time with $\mathbb {E}\left[T\right]<\infty$ such that $W_{T}$ and $Y$ have the same law. (We do not necessarily assume that $T$ was constructed by the method of Section 15.2.) Let $S_{K}=\inf \left\{ t:\left|W_{t}\right| \geq K\right\}$ . Prove that $T \leq S_{K}$ , a.s.

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Problem 15.7

(1)(2)(3)

Let $Y_{i}$ be a sequence of i.i.d. random variables with $\mathbb {P}\left(Y_{i}=1\right)=\mathbb {P}\left(Y_{i}=-1\right)=\frac{1}{2}$ , and let $S_{n}=\sum_{i=1}^{n} Y_{i} . S_{n}$ is called a simple symmetric random walk. Let $T_{1}, T_{2}, \ldots$ and $U_{1}, U_{2}, \ldots$ be as in Section 15.3.

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(1)

Prove that $\mathbb {E}\left[T_{1}^{p}\right]<\infty$ for all $p \geq 1$ .

(2)

Prove that if $\varepsilon >0$ ,

\lim _{n \rightarrow \infty } \frac{\sup _{k \leq n}\left|U_{k}-k\right|}{n^{(1 / 2)+\varepsilon }}=0, \quad \text{ a.s. }

Hint: Use Doob's inequalities to estimate

\mathbb {P}\left(\sup _{k \leq n}\left|U_{k}-k\right| \geq \delta n^{(1 / 2)+\varepsilon }\right)

(3)

Show that

\sup _{i \leq n}\left|W_{U_{i}}-W_{i}\right| / n^{(1 / 4)+(\varepsilon / 2)}

tends to zero in probability as $n \rightarrow \infty$ .

Problem 15.8

Let $S_{n}, T_{i}$ , and $U_{i}$ be as in Exercise 15.7. Prove that

\lim _{n \rightarrow \infty } \frac{\sup _{i \leq n}\left|W_{U_{i}}-W_{i}\right|}{\sqrt{n}}=0, \quad \text{ a.s. }

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Problem 15.9

Let $S_{n}$ be a simple symmetric random walk; see Exercise 15.7. Let $Y$ be a bounded symmetric random variable that takes values only in $\mathbb {Z}$ . ( $Y$ being symmetric means that $Y$ and $-Y$ have the same law.) Does there necessarily exist a stopping time $N$ such that $S_{N}$ and $Y$ have the same law? Why or why not?

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