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Math 635 -- Spring 2012\\[10pt]{\textbf{Homework 6\\[15pt]}}
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\noindent{\textbf {Due:  April 18th, 2012}, beginning of the class. Late homework will \textbf{not} be accepted. }\\






\begin{enumerate}
\item \begin{enumerate}
\item
Suppose that $X(t)=(X_1(t),X_2(t))\in \R^2$ satisfies
\begin{eqnarray*}
dX_1&=& 
-X_2 dB_t \\
dX_2 &=& X_1 dB_t
\end{eqnarray*}
and $|X(0)|=1$. Show that $X(t)$
does \emph{not} stay on the circle 
$|X| = 1$, even though the vector field 
$( 
-X_2 , X_1 )$ is tangent to the circle at $(X_1 , X_2 )$.\\(Hint: if $X(t)$ stays on the unit circle, is there a function that should remain constant?) 
\item  Show that if  $X(t)=(X_1(t),X_2(t))$ satisfies
\begin{eqnarray*}
dX_1 &=& 
-X_2 dB_t - 
\frac12 X_1 dt \\ 
dX_2 &=& X_1 dB_t 
-
\frac12 X_2 dt 
\end{eqnarray*}
with $|X(0)|=1$ then $X(t)$
stays on the unit circle. 
\end{enumerate}

\item Show that if $B_t$ is a standard BM then $M_t=(B_t + t) \exp(-B_t - t/2)$ is a martingale. 

\item Exercise 8.2 (integration by parts).  Use It\^o's formula to prove that if $h \in C^1(\R^+)$, then
\[
	\int_0^t h(s) dB(s) = h(t) B_t - \int_0^t h'(s)B_s ds.  
\]
Note, do not use formula (8.31).  Instead, I want you to verify the result directly using It\^o.
\end{enumerate}




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