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\title{Stochastic Processes}
\author{Summer Semester 2026, Exercise 5}
\date{Due Thursday, June 18, 2026}
\maketitle


\begin{enumerate}

\item Let $\eta_i$, $i=1,2, \dots$ be a sequence of independent random
variables with $P(\eta_i=1) = p$ and $P(\eta_i=-1)=q \equiv 1-p$.  Set
\[
  \xi_n = \eta_1 + \dots + \eta_n
\]
and let $\mathcal F_1 \subset \mathcal F_2 \subset \dots$ be the
natural filtration.
\begin{enumerate}
\item Prove that
\[
  \zeta_n = \Bigl( \frac{q}p \Bigr)^{\xi_n}
\]
is a martingale with respect to the same filtration.

\item Given $\lambda>0$, let
\[
  \theta_n = C^n \, \lambda^{\xi_n} \,.
\]
Determine the constant $C$ such that $\theta_n$ is a martingale with
respect to the same filtration.
\end{enumerate}

\item A gambler wins or looses one euro in each independent round of a
sequence of fair games.  He stops the game when he has won $a$ euros
or lost $b$ euros.  His initial balance is zero.

\begin{enumerate}
\item What is the probability of stopping the game with a positive
balance?

\item What is the expected stopping time?
\end{enumerate}

\emph{Note:} This is very close to the example given in class, with
slightly different parameters.  You should base your answer on the
optional stopping theorem.  You do not need to verify that the
conditions for the optional stopping theorem are met (this is
technical and the proof from class applies almost literally).

\item Repeat the previous problem when the probability of winning in
each round is $p$ and the probability of losing in each round is
$q=1-p$.

\emph{Hint:} Turn the process into a martingale using the trick from
Problem~1, then apply optional stopping.


\item Let $(\Omega, \mathcal F, P)$ be a probability space, $\mathcal
F_1 \subset \mathcal F_2 \subset \dots \subset \mathcal F$ a
filtration, and $\eta_1, \eta_2, \dots$ a sequence of integrable
random variables adapted to the filtration.  Assume that there exist
numbers $a_n$ and $b_n$ such that
\[
  \mathbb E[\eta_{n+1} | \mathcal F_n] = a_n \, \eta_n + b_n \,.
\]
Find two sequences $c_n$ and $d_n$ such that
\[
  \xi_n = c_n \, \eta_n + d_n
\]
is a martingale with respect to the same filtration.

\item $N$ balls are placed randomly in $K$ urns.  At each time step,
we choose one of the balls uniformly at random and place it into one
of the urns uniformly at random (so it is possible that a ball is
picked up and replaced into the same urn).  Let $\eta_n$ denote the
number of balls in the first urn at the end of time step $n$ of this
process.  We consider $\eta_n$ as a stochastic process with respect to
its natural filtration.

Use the construction from the previous problem to determine a process
\[
  \xi_n = c_n \, \eta_n + d_n
\]
that is a martingale with respect to the same filtration.


\end{enumerate}
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