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Math 635 -- Spring 2012\\[10pt]{\textbf{Homework 1\\[15pt]}}
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\noindent{\textbf {Due:  February 6th, 2012}, beginning of the class. Late homework will \textbf{not} be accepted. }\\



\begin{enumerate}

\item Exercise 1.3 from Steele (page 10).  Consider a simple random walk beginning at 0 and show that for any $k \ne 0$ the expected number of visits to level $k$ before returning to 0 is exactly 1.  Anyone who wants a hint might consider the number $N_k$ of visits to level $k$ before the first return to 0.  We have $N_0 = 1$ and can use the results on hitting probabilities to show that for all $k \ge 1$ we have
\begin{equation*}
	P(N_k>0) = \frac{1}{2k} \quad \text{and} \quad P(N_k>j+1|N_k>j) = \frac{1}{2} + \frac{1}{2} \frac{k-1}{k}.
\end{equation*}


\item Suppose that $X_i$ are i.i.d  random variables with $\E X_i=0, \E X_i^2=1, \E X_i^3=0$ and let $S_n=\sum_{i=1}^n X_i$. Find a cubic polynomial $f_n(x)$ (with $n$-dependent coefficients) such that $\{f_n(S_n)\}$ is a martingale with respect to $\{X_n\}$.

\item Suppose that $\tau_1$ and $\tau_2$ are stopping times with respect to $\{\FF_n\}$. Show that $\tau_1+\tau_2$ is also a stopping time.

\item This is a variant of Exercise 2.1 from Steele (page 27).  Consider a gambling game with multiple payouts: the player loses \$1 with probability $\alpha$, wins \$1 with probability $\beta$, and wins \$2 with probability $\gamma$.  Specifically, we assume that $\alpha = 0.52$, $\beta = 0.45$, and $\gamma = 0.03$, so the expected value of each round of the game is only \$-0.01.

\begin{enumerate}[(a)]
\item	Suppose the gambler bets one dollar on each round of the game and that $\{X_k\}$ is the amount won or lost on the $k$th round.  Find two real numbers $x_1,x_2$ such that $M_n = x_i^{S_n}$ is a martingale for each of $i \in \{1,2\}$  where $S_n = X_1 + X_2 + \cdots + X_n$ tallies the gambler's winnings.   Note: you will need to find the numerical solutions to a cubic equation but $x = 1$ is one solution so the cubic can be reduced to a quadratic.

\item Let $p$ denote the probability of winning \$100 or more before losing \$100.  Using \textit{both} martingales from part (a), find $p$.   To do so, you will need to use a high precision software package like Maple or Mathematica.  You should be sure to take proper account of the fact that the gambler's fortune may skip over \$100 if a win of \$2 takes place when the gambler's fortune is \$99.  
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