Math Problems Vault

Problem 11.1

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

\int _{0}^{t} \; \mathbb {1}_{\left\{ a\right\} }\left(W_{s}\right) d s=0, \quad \text{ a.s. }

for all $t>0$ .

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Problem 11.2

Let $a<b$ and let $f_{a, b}$ be the $C^{1}$ function such that $f_{a, b}(0)=f_{a, b}^{\prime }(0)=0$ and

f_{a, b}^{\prime }(x)=\int _{0}^{x} \; \mathbb {1}_{[a, b]}(y) d y, \quad x \in \mathbb {R}

In other words, $f_{a, b}$ is the function whose second derivative is $\; \mathbb {1}_{[a, b]}$ , except that the second derivative is not defined at $a$ and $b$ . Show Itô's formula holds for $f_{a, b}$ :

f_{a, b}\left(W_{t}\right)=\int _{0}^{t} f_{a, b}^{\prime }\left(W_{s}\right) d W_{s}+\frac{1}{2} \int _{0}^{t} \; \mathbb {1}_{[a, b]}\left(W_{s}\right) d s

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Problem 11.3

If $W_{t}$ is a Brownian motion, $a>0$ , and $T=\inf \left\{ t>0:\left|W_{t}\right|=a\right\}$ , calculate $\mathbb {E}\left[\int_{0}^{T}\left(W_{s}\right)^{k} d s\right]$ for each non-negative integer $k$ . Also calculate

\mathbb {E}\left[\int _{0}^{T} \; \mathbb {1}_{\left[b_{1}, b_{2}\right]}\left(W_{s}\right) d s\right]

if $\left[b_{1}, b_{2}\right] \subset [-a, a]$ .

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Problem 11.4

Let $W$ be a Brownian motion, let $t_{0}<t_{1}<\cdots <t_{n}=1$ , and let

B_{i}=\left(W_{t_{i}}-W_{t_{i-1}}\right)^{2}-\left(t_{i}-t_{i-1}\right) .

Show there exists a constant $c_{1}$ not depending on $\left\{ t_{0}, \ldots , t_{n}\right\}$ such that

\mathbb {E}\left[\sum _{i=1}^{n} B_{i}\right]^{2} \leq c_{1} \max _{1 \leq i \leq n}\left|t_{i}-t_{i-1}\right|

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Problem 11.5

Use Exercise 11.4 and the Borel-Cantelli lemma to prove that if $W$ is a Brownian motion, then

\lim _{n \rightarrow \infty } \sum _{k=1}^{2^{n}}\left(W_{k / 2^{n}}-W_{(k-1) / 2^{n}}\right)^{2}=1, \quad \text{ a.s. }

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Problem 11.6

(1)(2)

In our proof of Itô's formula, the use of stopping times simplifies the proof considerably. This exercise considers a proof of Itô's formula using fixed times. Suppose $M$ is a bounded continuous martingale, $A$ is a continuous process whose paths have total variation bounded by $N>0$ , a.s., and $X_{t}=M_{t}+A_{t}$ .

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(1)

Writing $[x]$ for the integer part of $x$ , prove that for each $t$ ,

\sum _{i=1}^{\left[2^{n} t\right]+1}\left(X_{(i+1) / 2^{n}}-X_{i / 2^{n}}\right)^{2}

converges in probability to $\langle X\rangle_{t}$ .

(2)

Prove that if $f$ is a $C^{2}$ function whose second derivative is bounded, then

\sum _{i=1}^{\left[2^{n} t\right]+1} f^{\prime \prime }\left(X_{i / 2^{n}}\right)\left(X_{(i+1) / 2^{n}}-X_{i / 2^{n}}\right)^{2}

converges in probability to

\int _{0}^{t} f^{\prime \prime }\left(X_{s}\right) d\left\langle X_{s}\right\rangle

Since the increments of $M$ and $A$ are not uniformly bounded by something small, this is much harder than the proof of Theorem 11.1 given in this chapter.

Problem 11.7

(1)(2)(3)

Here is an alternate way to prove Itô's formula.

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(1)

Suppose $X=M+A$ , where $M$ and $A$ are as in Exercise 11.6. Write

\begin{aligned} X_{t}^{2}-X_{0}^{2} & =\sum _{i=0}^{\left[t 2^{n}\right]-1}\left(X_{(i+1) / 2^{n}}^{2}-X_{i / 2^{n}}^{2}\right) \\ & =\sum _{i=0}^{\left[t 2^{n}\right]-1} 2 X_{i / 2^{n}}\left(X_{(i+1) / 2^{n}}-X_{i / 2^{n}}\right)+\sum _{i=0}^{\left[t 2^{n}\right]-1}\left(X_{(i+1) / 2^{n}}-X_{i / 2^{n}}\right)^{2} \end{aligned}

Use Exercise 11.6 to show that Itô's formula holds when $f(x)=x^{2}$ .

(2)

Derive the Itô product formula. Then use induction to show that Itô's formula holds when $f(x)=x^{n}, n$ a positive integer.

(3)

Given $f \in C^{2}$ , find polynomials $P_{m}$ such that $P_{m}, P_{m}^{\prime }, P_{m}^{\prime \prime }$ converge uniformly to $f, f^{\prime }, f^{\prime \prime }$ , respectively, on a compact interval as $m \rightarrow \infty$ . Apply Itô's formula for $P_{m}$ and show that one can take limits to derive Itô's formula for $f$ .