\documentclass[leqno, letterpaper, 11pt]{article}
\usepackage{amsmath, amsthm, amssymb,amsfonts, graphicx, verbatim, color}
\setlength{\textwidth}{17cm}
\setlength{\textheight}{23cm}
\setlength{\oddsidemargin}{0cm}
\setlength{\evensidemargin}{0cm}
\setlength{\topmargin}{0cm}
\setlength{\headheight}{0cm}
\setlength{\headsep}{0cm}
\setlength{\topskip}{0cm}
\setlength{\parskip}{0ex}


\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}
%\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bT}{{\mathbb T}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cA}{{\mathcal A}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{\mathcal H}
\newcommand{\cU}{\mathcal U}
\newcommand{\cY}{{\mathcal Y}}
\newcommand{\cZ}{{\mathcal Z}}
\newcommand{\cR}{{\mathcal R}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cW}{{\mathcal W}}
\newcommand{\cM}{{\mathbb M}}
\newcommand{\cO}{\mathcal O}
\newcommand{\cP}{\mathcal P}
\newcommand{\cT}{\mathcal T}
\newcommand{\cB}{\mathcal B}
\newcommand{\cS}{\mathcal S}
\newcommand{\cE}{\mathcal E}
\newcommand{\cQ}{\mathcal Q}
\newcommand{\cK}{\mathcal K}

\newcommand{\tS}{\tilde S}
\newcommand{\bS}{\bar S}
\newcommand{\tQ}{\tilde Q}
\newcommand{\bQ}{\bar Q}



\begin{document}

\null\vskip1cm


\noindent Math 236A, Fall 2012. 

\vskip0.5cm

\centerline{\bf Homework Assignment 3}

\vskip 0.5cm

\noindent {\bf Due:} Oct.~26, 2012

\vskip0.5cm

 
\noindent 1.  Let $\tau_a$ be the first time a standard Brownian motion in one 
dimension hits $a>0$. 

\noindent (a) Compute the density of $\tau_a$. 

\noindent (b) Either from (a) or (much more easily) by using an appropriate 
martingale, compute $E(\exp(-\lambda\tau_a))$, for any $\lambda>0$.

\noindent (c) Let $b>0$ and let $\tau_{-b}$ be the first time the 
Brownian motion hits $-b$. Show that  
$$
E(\exp(-\lambda\tau_{-b})1_{\{\tau_a<\tau_{-b}\}})=
E(\exp(-\lambda\tau_{a})1_{\{\tau_a<\tau_{-b}\}})\cdot
E(\exp(-\lambda\tau_{a+b})).
$$

\noindent (d) Let $\tau=\tau_a\wedge\tau_{-a}$. Compute $E(\exp(-\lambda\tau))$.



\vskip 0.5cm

\noindent 2. Let $B$ be the Brownian motion in two dimensions, started at $(0,a)$, $a>0$. Let 
now $\tau$ be the first time $B$ hits the line $\alpha x$. Also, let $X$ be the $x$-coordinate 
of the point $B(\tau)$. 

\noindent (a) Determine the density of $X$ when $\alpha=0$. ({\it Hint\/}. 
Condition on the value of the stopping time $\tau_a$ from problem 1(a).) 

\noindent (b) Now determine the density of $X$ when $\alpha\ne 0$. 
({\it Hint\/}. Use orthogonal invariance.) 

 

\end{document}
