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\noindent Fakult"at f"ur Mathematik,
Institut f"ur Mathematische Stochastik\\
Prof. G. Christoph, Lora Todorova%\\[1ex]
\begin{center}
{\bf Tutorial Stochastic Processes \quad
(Series 3, 2011/2012)}%\\[1ex]
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%Problems with * are additional problems.
\begin{enumerate}

%
\item[11.)] Let $N_t, t \geq 0$ be a homogeneous Poisson process
with intensity $\lambda >0$ and $\tau$ the time till the first
event. For $ s < t$ find  $P(\tau \leq s | N(t)=1)$! 
\item[12.)]
Suppose that $\{N_1(t), t\geq 0\}$ and $\{N_2(t), t\geq 0\}$ are
independent Poisson processes with rates $\lambda_1$ and
$\lambda_2$. $\{N_1(t)+ N_2(t), t\geq 0\}$  is a Poisson process
with rate $\lambda_1 + \lambda_2$. (Show it!). Prove that the
probability that the first event of the combined process
$\{N_1(t)+ N_2(t), t\geq 0\}$ comes from the first one $\{N_1(t),
t\geq 0\}$ is $\lambda_1\,/\,(\lambda_1 + \lambda_2)\,!$
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\item[13.)] 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[14.)]
 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[15.)] 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.   
\end{enumerate}

\end{document}
\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.