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Stochastic Calculus Final Exam \\
December 19, 2002
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\begin{itemize}
\item Answer all questions in the blue answer books.
\item On each book, write your name and the number of books used.  Number
each book, e.g.\ ``book 2 of 3''.
\item Cross out or erase all work you know to be wrong.  If you have a right
and a wrong answer, I will take off for the wrong answer as well as giving
credit for the correct one.
\item Explain and show all work, as time allows.  I will take off points
for formulas that are not explained.
\end{itemize}

\begin{description}
\item[1.] In each case, indicate whether the statement is true or false and
give a short explination of your answer, with either a reason for it to be
true or a counterexample.
\begin{description}
\item[a.] The random variables $X$ and $Y$ are jointly normal with mean
zero, $\mbox{var}(X) = 1$, $\mbox{var}(Y) = 1$, and 
$\mbox{cov}(X,Y) = \frac{1}{2}$.  Define $U=X-2Y$.  This $U$ is uncorrelated
with $X$, and independent of $X$.
\item[b.] With $X$ and $Y$ as above, define $V=X^2 + X - 2Y$.  This $V$ is
uncorrelated with $X$ and independent of $X$.
\item[c.] Let $X_n$ be a discrete time martingale.  Let ${\cal G}_n$ be the 
$\sigma-$algebra generated by $X_n$ only (i.e.\ not $X_{n-1}$, etc.).
Then $E[X_{n+1} \mid {\cal G}_n] = X_n$.
\item[d.] If $Y_n$ has $E[Y_{n+1}\mid {\cal G}_n ] = Y_n$, then $Y_n$
is a martingale.
\item[e.] If $\cal F$ and $\cal G$ are two algebras of sets in the
probability space $\Omega$, then
the intersection ${\cal F}\cap {\cal G}$ is also an algebra of sets.
\item[f.] If $Q$ is a probability measure on the space of continuous paths
$C([0,T]\to R)$ that is absolutely continuous with respect to Brownian
motion measure (Wiener measure), then the total variation of a path $X_t$ 
for $0\leq t \leq T$ is infinite with $Q$ probability one.  More formally,
if $A \subset C([0,T]\to R)$ is the set of paths with infinite total variation,
then $Q(A) = 0$ if $Q$ is absolutely continuous with respect to Wiener 
measure.
\end{description}

\item[2.] 
\begin{description}
\item[a.] For a normal random variable $X \sim {\cal N}(0,\sigma^2)$,
write the PDF for $X$ and use it to calculate $E[X^2 e^X]$. 
\item[b.] For $X$ and $Y$ as in question 1a, calculate $E[Y^2e^X]$.
Hint: express $X$ and $Y$ in terms of independent gaussians.
\end{description} 

\item[3.]
An $n$ state Markov chain has transition matrix $P$. The state after $t$ 
transitions is $X_t$, which is one of the values $(1,\ldots,n)$.  
We also have a Markov stopping time, $\tau\geq 0$, independent of the Markov 
chain.  A Markov stopping time depends on the parameter, 
$q = \mbox{Pr}(\tau > t \mid \tau \geq t)$.  
\begin{description}
\item[a.] Write an expression for $\mbox{Pr}(\tau = t)$.  
\item[b.] Find a matrix equation involving $P$ for the $\tau$ step
transition probabilities $v_{jk} = \mbox{Pr}(X_\tau = k \mid X_0 = j)$.
These $v_{jk}$ form the $\tau$ step transition probability matrix
$V$.  Write a matrix equation involving $V$ and $P$.
Hint: Starting from state $j$, break the event 
$A_k = \left\{X_{\tau}=k\right\}$ into
two sets, one with $\tau = 0$ ($Z = \left\{\tau = 0\right\}$)
and one with $\tau > 0$ ($B = \left\{\tau > 0 \right\}$).  It is easy
to calculate $\mbox{Pr}(X_{\tau} =k \mid Z, \; X_0=j)$.  
For the $B$ part, we can calculate
$$
\mbox{Pr}(X_{\tau}=k \mid B,\; X_0=j) = \sum_l 
\mbox{Pr}(X_{\tau}=k \mid X_1=l)\cdot \mbox{Pr}(X_1=l \mid B,\;X_0=j) \; .
$$
Because $\tau$ and $X_t$ are Markov, the first factors in the sum on the 
right are entries of $V$. 
\end{description}  

\item[4.] Suppose $x=(x_1,x_2)^* \in R^2$ and $f = (f_1,f_2)^* \in R^2$.  The
${}^*$ means transpose, so that the column vector $x$ has $x^*=(x_1,x_2)$,
which is a row vector.  The 
component of $f$ perpendicular to $x$ is 
$g = f - (x^*f/x^*x) x = f - (x^*f/r^2) f$, where 
$r^2 = \left\|x\right\|^2 = x_1^2 + x_2^2$.  The 
quantity in parentheses is a number, not a vector.  
\begin{description}
\item[a.] Show that $x^*g = 0$.
\item[b.] Show that for any $f(x)$, the system of two ordinary differential
equations 
$$
\frac{dx}{dt} = g(x) = f(x) - \frac{x^*f(x)}{r^2} x 
$$
has $r^2 = \mbox{\em const}$ by calculating
$$
\frac{d}{dt} x^*(t) x(t) = 2 x^* \frac{dx}{dt} \;\;\;\; \mbox{or} \;\;\;\;
d(r^2) = d(x^*x) = 2x^*dx = 2x^*gdt \; .
$$
\item[c.]
Suppose we have the two dimensional Ito SDE
$$
dX(t) = dW(t) - \frac{X(t)^* dW(t)}{R(t)^2} X(t) 
\;\;\;\;\mbox{with } \;\; R(t)^2 = X_1(t)^2 + X_2(t)^2\; .
$$
Here $X_1(t)$ and $X_2(t)$ refer to the first and second components of 
$X(t) \in R^2$.  Show that $R(t)$ is not constant in time.
\item[d.] Find a formula for $R(t)$ in terms of $R(0)$ and $t$.  Show that
if $R(0)$ is known, then $R(t)$ is not random.
\end{description}

\vfill
\pagebreak

\item[5.]
We have a system of two SDEs:
\begin{eqnarray*}
dS_t & = & rS_t dt + \sqrt{V_t} S_t dW_t^{(1)} \; , \\
dV_t & = & (a-bV_t) dt + \mu dW_t^{(2)} \; ,
\end{eqnarray*}
where $W^{(1)}$ and $W^{(2)}$ are independent Brownian motions.
The initial conditions for the SDE are $S_0 = 1$, $V_0 = .4$.  We want to 
calculate 
$$
f = E\left[\int_0^T S_t^2 dt \right] \; .
$$
Explain how to do this using a PDE.  Give the PDE and whatever other 
information (possibly including initial data, final data boundary conditions, 
etc.) is needed.  Make sure to give a probabilistic definition of the 
quantity that satisfies the PDE.


\end{description}

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