\documentclass[11pt,a4paper]{article}
\usepackage[ansinew]{inputenc}
\usepackage{ngerman}
\usepackage{latexsym}
\usepackage{amssymb,color}
\begin{document}
\thispagestyle{empty}
\noindent Fakult"at f"ur Mathematik,
Institut f"ur Mathematische Stochastik\\
Prof. G. Christoph%\\[1ex]
\begin{center}
{\bf Tutorial Stochastic Processes \quad
(Series 3, 2013/2014)}%\\[1ex]
\end{center}
%Problems with * are additional problems.
\begin{enumerate}

%

%
%
\item[12.)] Let $W(t)$ be a standard Wiener process. Using the fact, that a Gaussian process 
is determined by its expectation and covariance functions, prove:  the following processes are standard Wiener processes, too
$W_1(t) = - W(t), \; t \in [0,\infty)$\,,\\[1ex] 
$W_2(t) = c\cdot W(t \cdot c^{-2}),\; c > 0,\; t \geq 0$\,,\\[1ex]  
$W_3(t) =  t\cdot W(1/t) \; \mbox{for } t > 0,\; (W_3(0) = 0),\;t 
\in [0,\infty)$ and \\[1ex] 
$W_4(t) = W(h-t) - W(h),\;  h \; \mbox{fixed},\; t \in [0, h]$.
%
%
\item[13.)]
 Let $W(t) $ be a  Wiener process, $L > 0$ a  constant and $A \sim N(\mu, \tau^2 )$ be a
random variable independent of $\{ W(t) \}_{t \in [0, \infty)}$.  Calculate mean vector and covariance function of the vectors $(X_k(s), X_k(t))$, $s<t$, where
\\[1ex]
a) $ X_1 (t) = W (t + L) - W(L) $, \quad 
b) $ X_2 (t) = W (t + L) - W(t) $ \\[1ex]
c) $ X_3 (t) = A \cdot t + W (t) $,  \quad
d) $ X_4 (t) = \left\{ \begin{array}{cl} 
(1-t) W \left( \frac{\displaystyle t}{\displaystyle 1-t} \right) & , \quad 0 \leq t < 1 \\[1ex]
0 & , \quad t \geq 1 \; . 
\end{array} \right.$
%
\item[14.)] Let   $(W_t$, $t \geq 0)$ be a standard Wiener Prozess, then $ B(t)=W_t - t\,W_1$ for $0 \leq t \leq 1$ is a {\it Brownian bridge}.\\[0.5ex]
a) Prove $ E\,B(t) = 0 $ and $ Cov_B(t,s) = \min\{t,s\}\,-\,t\,s$.\\[0.5ex]
b) Prove, that the Process $X_t = (1+t)\,B(\frac{t}{1+t})$ for $0\leq t<\infty$ is a  Wiener process, too.   
%
\item[15.] Let $W_t$, $t \geq 0$ be a standard Wiener process. Define $X_t=  4\,W_t - t\,  W_4$, $0 \leq t \leq 4$.  Calculate $Cov(X_s,X_t)$ for $0\leq s,t \leq 4$.
%
\end{enumerate}

\end{document}
\item[16.)*]  Let $N_t, t \geq 0$ be a homogeneous Poisson process with intensity $\lambda >0$. 
how that $N(t)$ is continuous in mean square, which is to say that
$$E \{(N(t+h)\,-\,N(t))^2\} \to 0  \quad \mbox{as} \quad h \to 0\,!$$

%
\item[17.)*] Let $N_t, t \geq 0$ be a homogeneous Poisson process with intensity $\lambda >0$. 
Show that $N(t)$  is continuous in probability, which is to say that
$$P \{|N(t+h)\,-\,N(t)| > \varepsilon\} \to 0  \quad \mbox{as} \quad h \to 0\,, \quad \mbox{for all} \, \varepsilon > 0\,!$$


\item[18.)*] Let $N_t, t \geq 0$ be a homogeneous Poisson process with intensity $\lambda >0$ and $T > 0$ a constant. 
Calculate $E\left(\frac{N(t)}{t} - \lambda\right)^2$
and prove the so-called Law of Large Numbers for the Poisson Process:
$$P\left(\left|\frac{N(t)}{t} - \lambda\right| > \varepsilon\right)\to 0 \quad \mbox{as} \quad t \to \infty\,!$$
%
\item[19.)*] Let $N_t, t \geq 0$ be a homogeneous Poisson process with intensity $\lambda >0$ and $T > 0$ a constant. Calculate expectation function
and the variance of the so-called sample mean
$M_T (w) = \frac{\displaystyle 1}{\displaystyle T} \int\limits^T_0 N (t, w) dt \; , \; w \in 
\Omeg
%\newpage
{\textsf{\large \textbf{Examination\ \ \ Stochastic Processes}}} \hfill February 17, 2012

\small{}Otto-von-Guericke-Universität Magdeburg \\
Fakult"at f"ur Mathematik, \quad Prof.~Dr.~G.~Christoph\\

\vspace{2mm}
{\bf Please note the following:}
\begin{itemize}
\item The exam consists of 6 problems. You do not have to solve the individual
  problems completely, partial solutions are also possible.  You should
  clearly display your approach and way to solution. 
\item You can reach a maximum of {\bf 50 points}. For passing the exam
  a total of {\bf 20 points} is sufficient.
\item You are allowed to use: Pocket calculators, text books,
  mathematical and/or statistical tables, manuscripts and notes from
  lectures and/or exercises. 
\end{itemize}
\hrule
%
\begin{enumerate} 
\item[1.)] (10 points) Suppose $X_1 \sim N(1,1)$, $X_2 \sim N(-3,4)$, $X_3 \sim N(2,9)$ and let $X_1, X_2, X_3$ be independent.
Define $Y_1 =3\,X_3\,+\,X_2\,-\,3$ and  $Y_2 = 6\, X_1\,-\,X_3$.\\[1ex]
a) Find $E(Y_2)$ and $Var(Y_1)$.\\[1ex]
b) Calculate the  covariance-matrix   $\Sigma_{(Y_1,Y_2)} = \Big( E[(Y_i-EY_i)(Y_j-EY_j)]\Big)_{i,j=1,2}$\,!\\
c) Calculate the correlation coefficient $\rho(Y_1,Y_2)$!\\[1ex]
d) Find the conditional expectation $E\big((Y_2)^2\,|\,\sigma(X_1,X_2)\big)$.
%
\item[2.)] (6 points) Let $N(t)$, $t \geq 0$ be a homogeneous Poisson process with intensity $\lambda > 0$. 
Suppose $t_1 \leq t \leq t_2$. Calculate $p_2 = P(N(t) =2\,|\,N(t_1)=0\,,\, N(t_2) = 4)$!
%%
%
\item[3.)] (6 points) Let $W_t$, $t \geq 0$ be a standard Wiener process. Define $X_t=  4\,W_t - t\,  W_4$, $0 \leq t \leq 4$.  Calculate $Cov(X_s,X_t)$ for $0\leq s,t \leq 4$.
%
%
\item[4.)] (8 points)   Let $Y_0, Y_1, Y_2, ...$ be independent   random variables with $E|Y_k| \leq \infty$ and %\\
$E(Y_k)=0$ for all $k\geq 0$.
Set $S_0=S_1=0$ and $S_n = \sum_{k=2}^n \Big(Y_{k-2}\,Y_k\Big)$\,, $n \geq 2$. Calculate $E(S_n)$, $n\geq 0$, and prove that $\{S_n\}_{n\geq0}$ is a martingale with respect to the natural 
filtration ${\cal F}_n= \sigma(Y_0,...Y_n)$, $n \geq 0$. \quad {\bf {Hint:}} Prove $E(S_{n+1}|{\cal F}_n) = S_n$.
%
\item[5.)] (10  points) Let $W_t$, $t \geq 0$ be a standard Wiener process and let $\mu, c, \sigma$ and $r$ be positive constants. Consider the so called Cox-Ingersoll-Ross process
$X_t$ given by 
 $$dX_t\,=\,c(\mu\,-\,X_t)\,dt + \sigma\,\sqrt{X_t}\,dW_t\,,\qquad X_0=1\, \quad \mbox{and} \quad Y_t = f(t, X_t)= e^{-\,r\,t + 2\, X_t \,-\,2}\,. $$ 
a) Find $dY_t$. (Hint: Use the Extension II of the Ito Lemma!)\\
b) Under which condition on the parameters $\mu, c, r$ and $\sigma$ the process $Y_t$ is a  martingale with respect to the natural filtration ${\cal F}_t = \sigma(W_u , u \leq t)$.



\item[6.)](10 points) The following statements are TRUE or FALSE.\\ So just answer
TRUE, since ... or FALSE, since ....  giving only a short explanation. 
\begin{enumerate}
\item The random variables $Y_1$ and $Y_2$ in {\bf Problem 1} are independent.
\item Let $N(t)$, $t \geq 0$ be a homogeneous Poisson process with intensity $\lambda > 0$. 
Suppose $t_1 \leq t \leq t_2$. Then $p_1 = P(N(t) =1\,|\,N(t_1)=2\,,\, N(t_2) = 4)=0\,.$
\item Let $W_t$, $t \geq 0$ be a  standard Wiener process. The process $Y_t=2W_t - 2\,t$, $t \geq 0$, is  a martingale with respect to the natural filtration ${\cal F}_t = \sigma(W_u , u \leq t)$.
\item The process $(S_n)_{n\geq 0}$ in {\bf Problem 4} is also a martingale, if $E(Y_k)=1$ for all $k\geq 0$.
\item Let $W_t, t \geq 0$ be a standard Wiener process. The process $X_t= (1/\sqrt{t})\,(W_t)^2\,$ for $t > 0$ and $X_0=0$ is  a Gaussian process. 

\end{enumerate}

\item[12.)] Let $N_t, t \geq 0$ be a homogeneous Poisson process
with intensity $\lambda >0$. Show that $N(t)$ is continuous in
mean square:
$$ E \{(N(t+h)\,-\,N(t))^2\} \to 0  \quad \mbox{as} \quad h \to 0\,$$
and continuous in probability:
$$ P \{|N(t+h)\,-\,N(t)| > \varepsilon\} \to 0  \quad \mbox{as} \quad h \to 0\,, \quad \mbox{for all} \quad \varepsilon > 0\,!$$ 
\item[13.)]Prove that for $Y \sim N(0, \tau^2 )$ we have $E|Y|^4 = 3\, \tau^4$. Hence for the Wiener process $W(t)$ we find   
$     E (\mid W(t)-W(s) \mid)^4 = 3\, \sigma^4\,(\mid t-s \mid )^2$.
%
\item[15.*] 

\item[16.*] Let $N_t, t \geq 0$ be a homogeneous Poisson process
with intensity $\lambda >0$ and $T > 0$ a constant. Calculate
$E\left(\frac{N(t)}{t} - \lambda\right)2$ and prove the so-called
Law of Large Numbers for the Poisson Process:
$$P\left(\left|\frac{N(t)}{t} - \lambda\right| > \varepsilon\right)\to 0 \quad \mbox{as} \quad t \to \infty\,!$$
%
\item[17.*] Let $N_t, t \geq 0$ be a homogeneous Poisson process
with intensity $\lambda >0$ and $T > 0$ a constant. Calculate
expectation function and the variance of the so-called sample mean
$M_T (w) = \frac{\displaystyle 1}{\displaystyle T} \int\limits^T_0
N (t, w) dt \; , \; w \in \Omega \,!$

\item[22.)]
Show the 0.5-self-similarity of the Wiener Processe:
$$ (\sqrt{T}\, W(t_1), ..., \sqrt{T}\,W(t_n)) =^d (W(Tt_1),...,W(Tt_n))\,.$$
$X=^d Y $ means that $X$ and $Y$ have the same distribution,\\ i.e. $P(X \leq x)=P(Y\leq x)$ for all $x \in I\!\!R^n$.\\[1ex]
Hint: Prove that the Gaussian processes $X_1(t) = \sqrt(T) \,W(t)$ and $X_2(t) = W(T\,t)$ have the same 
expectation and covariance functions.