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\begin{document}

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\noindent Math 236A, Fall 2012. 

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\centerline{\bf Homework Assignment 4}

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\noindent {\bf Due:} Nov.~9, 2012

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\noindent 1. Show that the following processes are 
martingales, and compute their variances.

\noindent (a) $X_t=\int_0^t|B_s|\,dB_s$

\noindent (b) $X_t=\int_0^t M_s\,dB_s$, where $M_s=\max\{B_u: 0\le u\le s\}$.

\noindent (c) $X_t=\int_0^t B_{s/2}\,dB_s$,

\noindent (d) $X_t=\int_0^t I_s \,dB_s$, where $I_s=\int_0^sB_u^2\,du$. 

\noindent (e) $X_t=\int_0^t B_s \bar B_s \,dB_s$, where $\bar B_s$ is a Brownian 
motion independent of $B_s$. 

\noindent (f) $X_t=\int_0^t B_{\xi s} \,dB_s$, where $\xi$ is a uniform 
random variable on $[0,1]$, independent of $B_s$.

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\noindent 2. Consider $\int_0^tB_{s+1}\, dB_s$, $0\le t\le 1$. 

\noindent (a) This is not an It\^o integral. Why?

\noindent (b) Still, one may hope that, as $n\to\infty$, the limit of 
the approximating sums $\sum_{i=0}^{n-1} B_{t_i+1}(B_{t_{i+1}}-B_{t_i})$
exists in an appropriate sense. Show that the limit indeed exists (state in what sense), express 
it with genuine It\^o integrals, and show that it results in a continuous process 
which is {\it not\/} a martingale. 

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\noindent 3. The figure depicts a realization of $B_{k/n}$ (blue) 
and $\sum_{i=0}^{k-1} B_{i/n}(B_{(i+1)/n}-B_{i/n})$ (red)
for a large $n$, and $0\le k/n\le 200$ (plotted against $k/n$ and interpolated). 
Explain the presence of almost linear segments
in the red curve and its much larger oscillations. 
\begin{center}
\includegraphics[trim=0cm 0cm 0cm 0cm, clip, height=2.5in]{ft200.eps}
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\noindent 4. Compute the following stochastic integrals. The answers may contain 
Riemann integrals. 

\noindent (a) $\int_0^t e^s\, dB_s$

\noindent (b) $\int_0^t B_s^2\, dB_s$

\noindent (c) $\int_0^t e^{B_s}\, dB_s$. 

\noindent (d) $\int_0^t sB_s\, dB_s$.

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\noindent 5. Fix $a>0$ and 
let $\tau$ be the first time that Brownian motion $B_t$ hits either $a$ or $-a$.
Compute, with careful proof:
$$
E\int_0^\tau B_s^2 \, ds .
$$
({\it Hint\/}: It\^o's formula.)

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