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\begin{document}

\title{Stochastic Processes}
\author{Mock Exam}
\date{July 6, 2026}
\maketitle
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\begin{enumerate}

\item Let $\xi_n$ denote the standard symmetric random walk, i.e., 
\[
  \xi_n = \eta_1 + \dots + \eta_n
\]
where the $\eta_i$ are independent random variables with
$P(\eta_i=1) = P(\eta_i=-1)=\tfrac12$.

Compute the following.
\begin{enumerate}
\item $P(\xi_4=3|\xi_0=1)$
\item $P(\xi_4=3|\xi_0=2)$
\hfill (5+5)
\end{enumerate}

\item On a probability space $(\Omega, \mathcal F, P)$ with
sub-sigma-algebra $\mathcal G \subset \mathcal F$, we may define the
conditional variance of a random variable $\xi$ as
\[
  \Var(\xi | \mathcal G)
  = \mathbb E
    \bigl[(\xi - \mathbb E[\xi|\mathcal G])^2|\mathcal G\bigr] \,.
\]
\begin{enumerate}
\item Prove that
\[
  \Var(\xi) = \mathbb E[\Var(\xi|\mathcal G)]
  + \Var(\mathbb E[\xi|\mathcal G]) \,.
\]
\item Let $\xi_i$ be i.i.d.\ random variables with
$\mathbb E[\xi_i] = \mu$ and $\Var(\xi_i) = \sigma^2$.  Let $\nu$ be
an independent positive-integer-valued random variable and set
\[
  \eta_\nu = \xi_1 + \dots + \xi_\nu \,.
\]
Prove that
\[
  \mathbb E[\eta_\nu] = \mu \, \mathbb E[\nu]
\]
(Hint: Find $\mathbb E[\eta_\nu|\nu]$ first!), then conclude
\[
  \Var(\eta_\nu) = \mu^2 \, \Var(\nu) + \sigma^2 \, \mathbb E[\nu]
\]
(Hint: use part (a)).
\hfill (5+5)
\end{enumerate}

\item Let $\xi_n$, $n \in \mathbb N$, be an arbitrary integrable
stochastic process adapted to a filtration $\mathcal F_n$.  For $n
\geq 1$, set
\begin{equation*}
  \eta_n = \xi_n - \mathbb E[\xi_n|\mathcal F_{n-1}] 
\end{equation*}
and further $\zeta_0=0$ and
\begin{equation*}
  \zeta_n = \sum_{k=1}^n \eta_k
\end{equation*}
for $n\geq 1$.

Show that $\zeta_n$ is a martingale with respect to the filtration
$\mathcal F_n$.
\hfill(5)

\item Consider a game where a fair coin is tossed repeatedly.
Gamblers can bet on predicting the next toss.  If they predict
correctly, they win twice their stake; if they mispredict, they lose
it.

A single gambler is betting on observing the sequence HTHTHT as
follows.  They enter the game at some round with a stake of 1, betting
on observing H.  If they win, they bet their winnings on T, the next
letter of the sequence; if they lose, they quit the game.  So long as
they are in the game, they will continue betting on each next letter,
doubling their stake with each round, until they have either quit or
have observed the entire sequence HTHTHT, at which point they will
also quit the game, keeping their winnings.

\begin{enumerate}
\item Argue that $\xi_n$, the process representing a gambler's
winnings or losses, is a martingale.
\item Now, at every round of the game, a new gambler enters, playing the
same strategy.  Argue that the process $\zeta_n$ of cumulated winnings
or losses of all gamblers is a martingale.
\item The game is stopped at a time $\tau$ when the sequence HTHTHT is
observed for the first time.  Argue that
\begin{equation*}
  \zeta_\tau = 2^6 + 2^4 + 2^2 - \tau \,.
\end{equation*}
\item Apply the optional stopping theorem to find the expected number
of rounds that need to be played before the sequence HTHTHT is
observed.

(You may assume that the technical conditions for optional stopping
are satisfied; extra credit if you can name and prove some or all of
these conditions.)
\end{enumerate}
\hfill(3+3+4+5)


\item Let $\xi_n$ be a Markov chain with states $s_1, \dots, s_6$ and
transition matrix
\begin{equation*}
  \renewcommand*{\arraystretch}{1.2}
  P = \begin{pmatrix}
        \frac14 & \frac12 & \frac14 & 1 & 0 & 0 \\
        \frac14 & \frac12 & 0 & 0 & 0 & 0 \\
        0 & 0 & \frac14 & 0 & \frac14 & 0 \\
        \frac12 & 0 & 0 & 0 & \frac14 & 0 \\
        0 & 0 & \frac12 & 0 & \frac14 & 0 \\
        0 & 0 & 0 & 0 & \frac14 & 1
      \end{pmatrix} \,.
\end{equation*}
\begin{enumerate}
\item Find all classes of states (transient states and closed
irreducible classes of recurrent states).

\item Find all stationary distributions.
\end{enumerate}
\hfill(5+5)


\end{enumerate}



\end{document}
