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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,
(Serie 4)}%\\[1ex]
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Problems with * are additional problems.
\begin{enumerate} 
\item[16.)] Suppose $\Omega = \{1,2,3,4,5,6\}$ with $P(\{i\}) = 1/6$ for $i=1,2,3,4,5,6$.\\
Defin $X(\omega) = \omega$ = face value on die with $P(X=i) = 1/6$ for $i=1,2,3,4,5,6$\\
and $Y(\omega) = \left\{\begin{array}{ll} 2& \mbox{if}\quad \omega \in \{1,3,6\}\\
3 & \mbox{if}\quad \omega \in \{2,5\}\\
5 & \mbox{if}\quad \omega =4 \end{array} \right.$. \quad
Find $Z=E(X|Y)$ !
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\item[17.)] a) Let $X$ be uniform on $(0\,,\,1]$, $C_1 = (0\,,\,1/2]$, $C_2 = (1/2\,,\,1]$,\\  and ${\cal F}_2= \sigma(C_1, C_2)$.
Find $V=E(X|{\cal F}_2)$!\\
b) Remember Example 3.2 with ${\cal F}_4= \sigma(A_1, A_2, A_3, A_4)$.\\ Here $C_1=A_1 \cup A_2, C_2=A_3 \cup A_4$\\ 
and  $Z(\omega) = E(X|{\cal F}_4)=\left\{\begin{array}{ll} 1/8& \mbox{if}\quad \omega \in A_1\\
3/8 & \mbox{if}\quad \omega \in A_2\\
5/8 & \mbox{if}\quad \omega \in A_3\\
7/8 & \mbox{if}\quad \omega \in A_4 \end{array} \right.$. Find $U=E(Z|{\cal F}_2)$ !
 %
\item[18.)] Let $(W(t), \; t\ge0) $ be a standard Wiener Process  and $(N(t),
  \; t\ge0) $ a homogenuos Poisson Process mit dem Parameter $\lambda >0$. 
Prove that\\[1ex]
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(a) $W^2(t) + 2\,W(t)- t$,\\[0.5ex]
(b) $W^3(t) \,-\,3\,t\,W(t)$,\\[0.5ex]
(c*) $\exp\{-\theta W(t)- \frac{1}{2}\theta^2 \, t \}$
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\hfill and \hfill
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(d) $N(t) -\lambda t$,\\[0.5ex]
(e*) $(N(t) -\lambda t)^2 -\lambda t$,\\[0.5ex]
(f*) $\exp\{-\theta N(t) +  \lambda t (1-e^{-\theta}) \}$
\end{minipage}\\[2ex]
are martingales with respect to its natural filtration\\[1ex]
${\cal{F}}_t={\cal{F}}(W(u), \; 0 \le u \le t)$  or
${\cal{F}}_t={\cal{F}}(N(u), \; 0 \le u \le t)$ .\\[1ex]
(g) Show that $W^3(t)$ and $N(t)$ do not be martingales. 

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\item[19.)] Let $(W(t), \; t\ge0) $ be a standard Wiener Process. Calculate\\
(4.1) $E\int_0^t\,\,W_s\,dW_s =0$  and\\
(4.2) $E(\int_0^T W_s\,dW_s)^2 = \int_0^T E(W^2_s)\,ds$


\item[20*.)] Try to find a stochastic process $A_t$ such that
 $(W_t^4 \,+\,A_t \,,\,{\cal F}_t)$ is a martingale with respect to the filtration
${\cal F}_s=\sigma(W_u, 0\leq u \leq s)$, $s \geq 0$.\\
{\bf Hint:} \, First calculate $E\Big(\big((W_t-W_s)+W_s\big)^4\,|\,{\cal F}_s)$ for $s < t$  \,.
\end{enumerate}
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