Math Problems Vault

Problem 13.9.1

Let $X$ and $Y$ be independent standard Wiener processes. Show that, with $R_{t}^{2} = X_{t}^{2}+Y_{t}^{2}$ ,

Z_{t} = \int _{0}^{t} \frac{X_{s}}{R_{s}} d X_{s}+\int _{0}^{t} \frac{Y_{s}}{R_{s}} d Y_{s}

is a Wiener process. [Hint: Use Theorem (13.8.11) and Example (12.7.10).] Hence show that the squared process $R^{2}$ , called the squared Bessel process, satisfies

R_{t}^{2} = 2 \int _{0}^{t} R_{s} d W_{s}+2 t

Generalize this conclusion to $n$ dimensions.

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Problem 13.9.2

Write down the SDE obtained via Itô's formula for the process $Y_{t} = W_{t}^{4}$ , and deduce that $\mathbb {E}\left[W_{t}^{4}\right] = 3 t^{2}$ .

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Problem 13.9.3

Show that $Y_{t} = t W_{t}$ is an Itô process, and write down the corresponding SDE.

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Problem 13.9.4

Wiener process on a circle. Let $Y_{t} = e^{i W_{t}}$ . Show that $Y = X_{1}+i X_{2}$ is a process on the unit circle satisfying

d X_{1} = -\frac{1}{2} X_{1} d t-X_{2} d W, \quad d X_{2} = -\frac{1}{2} X_{2} d t+X_{1} d W

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Problem 13.9.5

(a)(b)(c)

Find the SDEs satisfied by the processes:

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(a)

$X_{t} = W_{t} /(1+t)$ ,

(b)

$X_{t} = \sin W_{t}$ ,

(c)

[Wiener process on an ellipse] $X_{t} = a \cos W_{t}, Y_{t} = b \sin W_{t}$ , where $a b \neq 0$ .

Problem 13.9.6

Find a random function $X(t)$ satisfying

d X(t) = \left(\frac{1}{2} X+\sqrt{1+X^{2}}\right) d t+\sqrt{1+X^{2}} d W(t)

with $X(0) = 0$ .

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