



\documentclass[a4paper,10pt]{article}

\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[geometry]{ifsym}
%\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
%\usepackage[finnish]{babel}
\usepackage[english]{babel}
\usepackage{graphicx}
\begin{document}

\newtheorem{lemma}{Lemma}
\newtheorem{thm}{Theorem}
\newtheorem{coro}{Corollary}
\newtheorem{example}{Example}
\newtheorem{ex}{Exercise}
\newtheorem{definition}{Definition}
\newcommand{\indep}{\perp\!\!\perp}
 \newcommand{\R}{\mathbb{R}} 
 \newcommand{\Q}{\mathbb{Q}} 
 \newcommand{\N}{\mathbb{N}} 
\newcommand{\F}{\mathbb{F}} 
 \newcommand{\C}{\mathbb{C}} 
\newcommand{\Z}{\mathbb{Z}} 
 \newcommand{\PP}{{\mathbf {P}}} 
\newcommand{\Tau}{{\mathcal {T}}} 

\date{\today}











%opening
%\title{}
%\author{}

{\bf Stochastic analysis, spring  2013, Exercises-5, 21.02.2013 }
\begin{enumerate}


\item
We have seen that when $E_P(|X|)<\infty$, 
\begin{align*}
\forall \varepsilon >0, \; \exists \delta : \; P(A)<\delta \Longrightarrow E_P\bigl( |X| {\bf 1}_A \bigr) < \varepsilon 
\end{align*}
Show that  a collection ${\mathcal C} \subset L^1(P)$ is uniformly integrable
if and only if 
\begin{align*}
\forall \varepsilon >0, \; \exists \delta : \; P(A)<\delta \Longrightarrow  \sup_{X\in {\mathcal C} } E_P\bigl( |X| {\bf 1}_A \bigr) < \varepsilon 
\end{align*}





\item Let $\tau(\omega)\in \N$ be a stopping time w.r.t. $\F=( {\mathcal F}_t: t\in \N )$. Show that
\begin{eqnarray*}
  {\mathcal F}_{\tau} = \bigl \{ A \in {\mathcal F}:   A\cap \{ \tau \le t \} \in {\mathcal F}_t \forall t \in \N \}
\end{eqnarray*}
is a $\sigma$-algebra.





\item We continue with the random walk. We have
\begin{eqnarray*}
 M_t(\omega)= \sum_{s=1}^t X_s(\omega)
\end{eqnarray*}
is a binary random walk where $t\in \N$ and
$(X_s: s\in \N) $ are i.i.d. random variables with
\begin{eqnarray*}
   P( X_s=\pm 1 ) = P( X_s= \pm 1 | {\mathcal F}_{s-1}   )=   1/2
\end{eqnarray*}
$X_s$ is ${\mathcal F}_s$ measurable and $P$-independent from ${\mathcal F}_{s-1}$.

Recall that $(M_t)_{t\in \N}$ and  $(M_t^2-t)_{t\in \N}$ are   $\F$-martingales.

\begin{itemize}

\item
Consider the stopping time $\tau=\tau_K = \inf\{ t : M_t \ge K \}$ for $K\in \N$.
Show that $P( \tau < \infty )= 1 $.

Hint: the stopped martingale $( M_{t\wedge \tau }  : t\in \N)$ is a sub-martingale
 bounded from above 

(equivalently $(-M_{t\wedge\tau} )$ is
a supermartingale bounded from below).

 Apply Doob forward convegence theorem,

\item Show that $P$ almost surely $M_{\tau}(\omega)=K$ 

\item Show that $( M_{t\wedge \tau }(\omega) : t\in \N )$ is not uniformly integrable.

Hint: otherwise we could interchange the expectation and  the limit for
 $t\to\infty$ operations.
 
 
\item Show that $E( \tau )= +\infty$

Hint: prove it by contradiction, using
 \begin{eqnarray*}
|M_{t\wedge \tau}(\omega) | \le t\wedge\tau (\omega) \le \tau(\omega) \quad \forall
t\in \N
\end{eqnarray*}
\end{itemize}

{\bf Resume }: a gambler plays a fair coin-toss game with unit stakes, playing
from time $0$ 
  until  the stopping time $\tau_K(\omega)$,  when he quits
the game   a profit $K>0$.
With probability one  $\tau_K(\omega)<\infty$, the gambler always
makes a  profit  $K$ which is arbitrarilty large.

This free-lunch paradox  is explained as follows:

The gambler's strategy, to play  until $\tau_K(\omega)$
requires an infinite amount of capital, because $\forall M\in \N$
$P( \tau_{-M} > \tau_K ) >0$,  for any finite amount of capital
there is a positive probability to lose everything before $\tau_K$.

And even with an infinite amount of capital at disposal, altough $\tau_K(\omega)$
is $P$ a.s.  finite,
the expected time  for winning $K$ is $E(\tau_K)=\infty$. 


\item
  A three-player ruin problem: Initially, three players have  respectively $a, b, c \in \N$
units of capital.
 Games are independent and each game consists of choosing two players at random
and transferring one unit from the first-chosen to the second-chosen player.
Once a player is ruined, he is ineligible for further play. 
\\

Let $\tau_1$ be the number
of games required for one player to be ruined, and let $\tau_2$  be the number of
games required for two players to be ruined.
\\



 Let $(X_t, Y_t, Z_t)$ be the numbers of units possessed by the three
players after the $t$-game, and
\begin{eqnarray*} &&
M_t := X_t Y_t Z_t +  \frac{ (a + b + c) t } 3   \quad\mbox{ and }  \\ &&
 N_t := X_t Y_t + X_t Z_t + Y_t Z_t + t
 \end{eqnarray*}


\begin{itemize}

\item
Show that the stopped processes $(M_{t \wedge \tau_1}: t\in \N)$ and 
$(N_{t \wedge \tau_2}: t\in \N)$ 
are non-negative $\F$-martingales where  ${\mathcal F}_t = \sigma(X_s,Y_s,Z_s, s\le t)$.

\item
Use Doob martingale convergence theorem  and Fatou lemma
to show  that $E(\tau_k) <  \infty$,
for $k=1,2$

\item   Knowing that  $E(\tau_k) <  \infty$, 

show that $(M_{t \wedge \tau_1}: t\in \N)$
and $(N_{t \wedge \tau_2}: t\in \N)$ are uniformly integrable.

\item Use uniform integrability of the stopped martingales  $(M_{t \wedge \tau_1}: t\in \N)$
and $(N_{t \wedge \tau_2}: t\in \N) $ to  compute $E(\tau_k)$ for $k=1,2$.

\end{itemize}




\item
A generalization of a game by Jacob Bernoulli.
In this game a fair die is rolled, and if the result is $Z_1$, then
$Z_1$ dice are rolled. If the total of the $Z_1$ dice is $Z_2$, then $Z_2$ dice are rolled. If
the total of the $Z_2$ dice is $Z_3$, then $Z_3$ dice are rolled, and so on. Let $Z_0\equiv 1$.

Find a positive constant $\alpha$ such that 
 \begin{eqnarray*}
M_t(\omega)=  Z_t (\omega) \alpha^t   \quad  t\in \N 
 \end{eqnarray*}
is a  $\F$-martingale  where ${\mathcal F}_t = \sigma(Z_0,Z_1, \dots,Z_t)$.

Hint: compute $E(Z_{t+1} |{\mathcal F}_t )$ 


What does Doob's martingale convergence theorem tell us about this?




\item 
\begin{itemize}
 \item 
If $(M_t(\omega):t\in \N)$ is a $\F$-martingale and $f(x)$ is convex
such that $E(|f(X_t)|)< \infty  \forall t\in \N$,
show that  $(f( M_t(\omega)):t\in \N)$  is an $\F$-submartingale.



\item
If $(M_t(\omega):t\in \N)$ is a $\F$-submartingale and $f(x)$ is convex non-decreasing
such that $E(|f(X_t)|)< \infty  \forall t\in \N$,
show that  $(f( M_t(\omega)):t\in \N)$  is an $\F$-submartingale.


Hint: use  Jensen inequality for conditional expectation.
  
\end{itemize}

\end{enumerate}

\end{document}
