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Math 635 -- Spring 2012\\[10pt]{\textbf{Homework 2\\[15pt]}}
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\noindent{\textbf {Due:  Friday, February 24, 2012}, beginning of the class. Late homework will \textbf{not} be accepted. }\\



\begin{enumerate}

\item Exercise 2.4 in text, page 28.  If $\{M_n, \F_n\}$ is a martingale with 
\[
\E [M_n^2] < \infty \quad \text{ for all } n \ge 0,
\]
show that we can write
\begin{equation*}
	M_n^2 = N_n + A_n,
\end{equation*}
where $(1)$ $\{N_n,\F_n\}$ is a martingale; $(2)$ $A_n$ is monotone (so, $A_n \ge A_{n-1}$); and $(3)$ $A_n$ is nonanticipating (so $A_n \in \F_{n-1}$).  Those who want a hint might try defining the $A_n$ by first taking $A_0 = 0$ and then setting 
\begin{equation*}
	A_{n+1} = A_n + \E\left[(M_{n+1} - M_n)^2 \ | \ \F_n\right], \text{ for } n \ge 0.
\end{equation*}

(Why may we care about something like this?  Note that it implies that if the increments, $(M_{n+1} - M_n)^2$ are ``small'', then $\E M_n^2$ is also small.  This is an example of using local dynamics (one time increment) to prove something about the global dynamics.)

\item Recall Theorem 2.8, the $L^1$ Bounded convergence theorem.  Here we will demonstrate that we need not have $X_n \to X_{\infty}$ in $L^1$.
\vspace{.1in}

Let $X_1,X_2, \dots$ be independent random variables with
\begin{equation*}
	P\left(X_i = \frac{3}{2}\right) = P\left(X_i = \frac{1}{2}\right) = \frac{1}{2}.
\end{equation*}
Let $M_0 = 1$, and for $n>0$ let $M_n = X_1\cdots X_n$.
\begin{enumerate}
	\item Show that $M_n$ is a martingale with respect to $\F_n = \{X_1,\dots,X_n\}$ and that it satisfies all the hypothesis of Theorem 2.8.
	\item By considering $\ln M_n$, us the law of large numbers to conclude that $M_n \to 0$ a.s. as $n \to \infty$.
	\item Conclude that $\|M_n - M_{\infty}\|_{1}$ does not converge to zero as $n \to \infty$.
	\item Explain why this example does not contradict the $L^2$ bounded convergence theorem.
\end{enumerate}
 
 

\item  For $a>0$, let 
\[
X_t = \frac{1}{\sqrt{a}} B_{at},
\]
where $B_t$ is a Brownian motion.  Show that $X_t$ is a Brownian motion.

\item Let $X_t$ and $Y_t$ be independent standard Brownian motions.  Show that 
\[
	Z_t = \frac{1}{\sqrt{2}} (X_t - Y_t)
\]
is also a standard Brownian motion.

 
\item \textbf{(Exercise 3.1)} Let $U_t$ be a standard Brownian bridge (see page 41 for the definition).
\begin{enumerate}
\item Show that we can write $U_t = B_t - tB_1$ for $0 \le t \le 1$.
\item Show that $Cov(U_s,U_t)=s(1-t)$ for $0\le s\le t\le 1$.
\item Let $X_t=g(t) B_{h(t)}$, and find functions $g$ and $h$ such that $X_t$ has the same covariance as the Brownian bridge. ($B_t$ is a standard BM.)  Hint: narrow your search by taking $g(t) = t$ and assume $h$ is monotone decreasing.
\item Show that $Y_t=(1+t) U_{t/(1+t)}$ is a BM on $[0,\infty)$.

\end{enumerate}

\end{enumerate}



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