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\begin{document}
\title{Mathematics 522: Test I}
\author{February 25, 1998}
\maketitle

{

\vspace{0.4in}
\noindent\large Name: 
............................................................. 
\vspace{0.3in}
}

Do Problems 6 and your choice of three of the remaining five problems. There are 
blank pages at the end if you need them. \bigskip


\begin{problem}Let $(\Omega,\sF,P)$ be a probability triple. \begin{enumerate}
\item State the Borel-Cantelli Lemmas; and \vspace{2.2in}
\item State one of the Fatou Lemmas.
\end{enumerate}
\end{problem}
\vspace{1.7in}

\begin{problem}Let $(\Omega,\sF,P)$ be a probability triple. Prove one of the 
Borel-Cantelli Lemmas or one of the Fatou Lemmas. \end{problem}
\vspace{2.5in}

\newpage

\begin{problem} Let $X_1,X_2,\ldots$ be a sequence of real valued random 
variables on a probability triple $(\Omega,\sF,P)$. Show that $P(\{X_n \mbox{\ 
\rm
converges}\})=0$ or $=1$. {\em (HINT: Point out why $\{X_n \mbox{\ \rm 
converges}\}$ is a tail event, and quote an appropriate result about tail 
algebras.)}
\end{problem}
\vspace{2.4in}


\begin{problem} Let $(\Omega,\sF,\{\sF_n\},P)$ be a filtered space. Let $X\in 
L^1(\Omega,\sF,P)$.
\begin{enumerate}
\item Give the definition of a martingale on $(\Omega,\sF,\{\sF_n\},P)$. 
\vspace{2in}
\item Letting $X_n:=E(X|\sF_n)$ for $n=0,1,\ldots$, prove that $X_0,X_1,\ldots$ 
is a martingale. \end{enumerate}
\end{problem}

\vspace{2in}
\newpage
\begin{problem} Let $(\Omega,\sF,\{\sF_n\},P)$ be a filtered space. Let 
$X_0,X_1,\ldots$ be real valued functions $\in L^1(\Omega,\sF,P)$. Assume that 
the process $X_0,X_1,\ldots$ is adapted to the filtration $\{\sF_n\}$. 
\begin{enumerate}
\item What does it mean
when we say that the stochastic process $X_0,X_1,\ldots$ is adapted to the 
filtration $\{\sF_n\}$? \vspace{1.7in}
\item Assume that $\displaystyle \lim_{i\to \infty}E(|X_i-2|)=0$, where $2$ 
denotes the constant function. Let $T:\Omega\to \zed^+$ denote the map which 
sends $\omega\in \Omega$ to the smallest integer $n$ such that $X_n(\omega)\in 
[1,4]$ or to $\infty$ if no $X_n(\omega)\in [1,4]$. Show that $T$ is an almost 
surely finite stopping time. \vspace{2.4in}
\item Show that $T$ is in fact almost surely bounded, i.e., that there exists a 
real number $K>0$ such that $T\le K$ off of a set of $P$-measure zero. 
\end{enumerate}
\end{problem}
\vspace{2.4in}

\newpage
\begin{problem}\label{last} Let $X_0,X_1,\ldots$ denote a martingale on a 
filtered space $(\Omega,\sF,\{\sF_n\},P)$. Let $T$ denote a stopping time on 
$(\Omega,\sF,\{\sF_n\},P)$.
\begin{enumerate}
\item Give the definition of the stopped process $X^T$; and \vspace{2.2in}
\item Show that $X^T$ is a martingale.
\end{enumerate}
\end{problem}

\vspace{2in}

\newpage
\ \ \
\newpage

\ \ \
\end{document}



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\begin{document}
\title{Mathematics 522: Test 2}
\author{April 8, 1998}
\maketitle

{

\vspace{0.4in}
\noindent\large Name: 
............................................................. 
\vspace{0.3in}
}

Do Problem 4 and your choice of three of the remaining five problems. There are 
blank pages at the end if you need them. \bigskip

\begin{problem}Let $X:=\{X_n,n=0, 1, 2, \dots\}$ be an adapted stochastic 
process on a filtered space $(\Omega,\sF,\{\sF_n\},P)$. \begin{enumerate}
\item State a result of Doob giving conditions on $\{X_n\}$ so that for $c>0$ we 
have a bound for $P(\{\sup_{n\le N} X_n\ge c\})$ in terms of $E(|X_N|)$.
\vspace{2.4in}
\item Assuming that $X$ is a martingale, 
state some conditions that guarantee that $X_\infty :=\lim_{n\to \infty} X_n$ 
exists almost surely and moreover that for all finite $n$, $X_n$ is a version of 
$E(X_\infty|\sF_n)$. \end{enumerate}
\end{problem}


\newpage
\begin{problem}Let $Y$ be a real valued random variable defined on a probability 
triple, $(\Omega,\sF,P)$.
\begin{enumerate}
\item If $Y$ is normal, then the moment generating function of $Y$, 
\begin{equation}\label{theFormula}
\large M_Y(t):= E(e^{tY}) = e^{\mu_Y t+\frac{\sigma_Y^2 t^2}{2}} \end{equation}
where the mean is $E(Y)=\mu_Y$ and $\sigma_Y^2$ is the variance $\displaystyle 
E((Y-\mu_Y)^2)=E(Y^2)-\mu_Y^2$ of $Y$. Show this under the assumptions that 
$\mu_Y=0$ and $\sigma_Y^2=1$. Note that in this case the probability density 
function $p_Y(x)$ of $Y$ is $$\large \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}.$$ 
\vspace{3.5in}
\item Assuming formula (\ref{theFormula}) above and the fact that ``if the 
moment generating function $M_Z(t)$ of a random variable $Z$ exists for $t$ in a 
neighborhood of $t=0$, then $M_Z(t)$ determines the probability density function 
of $Z$,'' show that a linear combination $\displaystyle c_1Y_1+\cdots+c_nY_n$ of
a finite number of independent normal random variables $Y_1,\ldots,Y_n$ is 
normal 
with mean $\displaystyle c_1\mu_{Y_1}+\cdots+c_n\mu_{Y_n}$ and variance 
$\displaystyle c_1^2\sigma^2_{Y_1}+\cdots+c_n^2\sigma^2_{Y_n}$. (Here 
$c_2,\ldots,c_n$ are real numbers.)
\end{enumerate}
\end{problem}
\vspace{3in}
\newpage


\begin{problem} Let $M_t$ be a right continuous $L^2$ martingale on 
$\displaystyle (\Omega,\sF,\{\sF_t;t\ge 0\},P)$ with right continuous paths.
\begin{enumerate}
\item Assuming that $\ t\ge s$, show that $\displaystyle E(M_t^2-
M_s^2|\sF_s)=E((M_t-M_s)^2|\sF_s)$. \vspace{3in}
\item Assuming that $\ t\ge s$ and that $F\in \sF_s$ what is the Dol\'eans 
measure of $F\times (s,t]$? \end{enumerate}
\end{problem}
\vspace{2.5in}

In what follows let $B_t, t\ge0$ denote the usual Brownian motion, i.e., 
\begin{enumerate}
\item $\Omega := C[0,\infty)$, the continuous real valued functions on 
$[0,\infty)$;
\item for $t>0$, $B_t$ denotes a real valued map on $ C[0,\infty)$ with $B_0=0$;
\item $\sF$ denotes the $\sigma$-algebra generated by the $B_t$; \item the 
probability measure (so called Wiener measure) $P$ on $\sF$ is the unique 
probability measure so that the probability density function of $B_t$ for $t>0$ 
is the normal density
$$\displaystyle \frac{1}{\sqrt{2\pi t}}\int_{-\infty}^xe^{-\frac{u^2}{2t}}du;$$ 
\item \label{missing} $B_t$ satisfies some conditions on the increments $B_s-
B_s$. \end{enumerate}

\newpage

\begin{problem}Assume you have a Brownian motion as above. \begin{enumerate}
\item State the exact properties alluded to in item \ref{missing} above. 
\vspace{2.7in}
\item Show that $B_t$ is
a martingale relative to the filtration $\displaystyle\sF_t:= \sigma(B_s; s\le 
t)$. \vspace{3in}
\item Show
that $B_t^2-t$ is a martingale relative to the filtration $\displaystyle\sF_t:= 
\sigma(B_s; s\le t)$. \end{enumerate}
\end{problem}
\vspace{3in}



\begin{problem}\label{prob2}Give arguments to show that \begin{enumerate}
\item $\displaystyle X_t:=-B_t$ is a standard Brownian motion; \vspace{2.5in}
\item for any $c>0$, $\displaystyle X_t:=\sqrt{c}B_{t/c}$ is a standard Brownian 
motion; and \vspace{2.5in}
\item $\displaystyle X_t:=tB_{1/t} {\rm \ for\ }t>0$ and $X_0:=0$ is a standard 
Brownian motion (do not worry about the continuity of paths at $t=0$). 
\end{enumerate}
\end{problem}

\newpage



\begin{problem} Assume that the above filtration has been completed so that 
$\sF_s = \cap_{t>s} \sF_t$.
\begin{enumerate}
\item Give the definition of an optional time $T$ ($T$ is also called a stopping 
time). \vspace{3.1in}
\item Show that
$\displaystyle T_c:= \inf \{t\in [0,\infty) | B_t=c\}$ 
is an optional time.
\end{enumerate}
\end{problem}
\vspace{2.4in}





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\begin{document}
\title{\vskip 2.5cm Mathematics 522: Final Exam} \author{Wednesday, May 6, 1998}
\maketitle


\vspace*{-1.5in}
\noindent{\large Name:
$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ }$}
\vspace*{1.5in}

\section*{Problems} Do four of the following problems. 


\begin{problem}Let $B_t$ denote a standard Brownian motion on $[0,\infty)$ with 
filtration on $C[0,\infty)$ given by ${\cal F}_t := \sigma\{B_s | s\le t\}$. 
\begin{enumerate}
\item Show that $B_t^2-t$ is a martingale relative to the ${\cal F}_t$. 
\vspace*{2.5in}
\item For $c>0$, let $T_c:=\inf\{t\ge0 | B_t\ge c\}$. Compute $E(T_c)$. 
\end{enumerate}
\end{problem}

\newpage



\begin{problem}Assume that $dS= 2 dt + t dB_t.$ Compute $dV $, where 
$V=t\sin(S)$, in terms of $dt$ and $dB_t$. \end{problem}


\vspace{2.2in}

\begin{problem}Let $B_t$ denote a standard Brownian motion on $[0,\infty)$ with 
filtration on $C[0,\infty)$ given by ${\cal F}_t := \sigma\{B_s | s\le t\}$. 
Compute
\begin{enumerate}
\item $\displaystyle \int_0^t dB_s$;
\vspace*{2.2in}
\item $\displaystyle \int_0^t B_sdB_s$.
\end{enumerate}
\end{problem}



\newpage


\begin{problem}\label{martingaleExponential}Let $h(s)$ be an element of 
$C[0,\infty)$. \begin{enumerate}
\item Show that
$$\displaystyle X_t = e^{\int_0^th(s)dB_s-\int_0^t\frac{h(s)^2}{2}ds}$$ 
satisfies
$\displaystyle dX_t=h(t)X_tdB_t.$
\vspace*{2.5in}
\item Why does this imply that $X_t$ is a martingale relative to the filtration 
${\cal F}_t$ of the preceding problem?
\end{enumerate}
\end{problem}




\vspace{2.5in}
\newpage

\begin{problem}
Let
$$N_t=N_0e^{rt-\frac{\sigma^2}{2}t+\sigma B_t}$$ 
where $N_0,r,\sigma$ are constants.
\begin{enumerate}
\item Show that $N_t$ satisfies
$\displaystyle dN_t=rN_tdt+\sigma N_tdB_t$. \vspace*{2.6in}
\item Show that $\displaystyle E(N_t) =N_0e^{rt}$. {\em Hint: You might find the 
martingale from Problem \ref{martingaleExponential} with $h(s)=\sigma$ useful.}
\end{enumerate}
\end{problem}
\newpage


\begin{problem}
For $0\le t<1$ let
$\displaystyle Y_t=t+(1-t)\int_0^t\frac{dB_s}{1-s}.$ \begin{enumerate}
\item Show that
$\displaystyle dY_t =\frac{1-Y_t}{1-t}dt+dB_t.$ \vspace*{3in}
\item Compute $E(Y_t)$ and $E([Y_t-t]^2)$. \end{enumerate}
\end{problem}



\vfill
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\Large \em Have a Happy Summer!\\
\ \ \\
\end{center}
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