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\begin{document}

\title{Numerical Methods II}
\author{Problem Set 8}
\date{due in class, May 12, 2004}
\maketitle

\begin{enumerate}

\item \textbf{Project:} Write an \textsf{Octave} program to verify
  experimentally that
\[
  \mathbb{E} \biggl| \sum_{i=0}^{n-1} (W_{i+1}-W_{i})^2 - T \biggr|^2
  \to 0 \quad \text{as } n \to \infty \,,
\]
where $W(t)$ is Brownian motion, $t_i = i \, \delta t$ with $\delta t
= T/n$, and $W_i=W(t_i)$.

\item Use the It\^o formula to show that the so-called \emph{It\^o
  equation}
\[
  \d X = \d t + 2 \sqrt X \, \d W 
\]
has the explicit solution
\[
  X(t) = \bigl( W(t) + \sqrt{X(0)} \bigr)^2 \,.
\]

\item \textbf{Project:} Write an \textsf{Octave} program to solve the
It\^o equation with deterministic initial data $X(0)=1$ with
Milstein's method.  Show that the method converges with strong order
1.

\end{enumerate}

\emph{Hint:} For both programming projects. you can use the programs
from Desmond Higham's article with minor modifications.



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