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Math 635 -- Spring 2012\\[10pt]{\textbf{Homework 3\\[15pt]}}
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\noindent{\textbf {Due:  March 12, 2012}, beginning of the class. Late homework will \textbf{not} be accepted. }\\



\begin{enumerate}
\item \textbf{(Exercise 4.1, a)}   (Tower property) Use the definition of conditional expectation to prove that if $\mathcal H$ is a sub-$\sigma$ field of $ \mathcal G$  then 
\[
\E(\E(X|\mathcal G)|\mathcal H)=\E(X|\mathcal H).
\]

\item \textbf{(Exercise 4.2, a)} Show that if $\E (|X_n - X|^{\alpha}) \to 0$ for some $\alpha > 0$, then $X_n$ converges to $X$ in probability. 

\item \textbf{(Exercise 4.5)} Suppose that $\tau$ is a stopping time for the filtration $\{\FF_n\}$ and that there is a constant $N$ such that for all $n \ge 0$ we have
\begin{equation}\label{eq:main1}
	P(\tau \le n + N \ | \ \FF_n) \ge \epsilon > 0.
\end{equation}
Informally, equation \eqref{eq:main1} tells us that - no matter what has happened so far - there is at least an $\epsilon$ chance that we will stop sometime in the next $N$ steps.

Use induction and the trivial relation
\[
	P(\tau> kN) = P(\tau > kN\text{ and } \tau > (k-1)N)
\]
to show that $P(\tau> kN) \le (1-\epsilon)^k$.  Conclude that we have $P(\tau < \infty) = 1$ and that $\E \tau^p < \infty$ for all $p \ge 1$.

Note: Here one should note that $P(A|\FF_n)$ is interpreted as $\E(1_A|\FF_n)$, and any honest calculation with $P(A|\FF_n)$ must rest on the properties of conditional expectations.

\item Show that with probability one the standard BM is not strictly increasing on any interval. 
\\
Hints: First show that it is enough to prove that $P(\textup{$B_t$ is strictly increasing on $[a,b]$})=0$ for any $0\le a<b$. Then use the fact that $$P(\textup{$B_t$ is strictly increasing on $[a,b]$})\le P(B_{t_1}<B_{t_2}<\dots B_{t_k}), \quad\textup{if}\quad a\le t_1<t_2<\dots\le b.$$
\end{enumerate}




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