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\begin{document}

\centerline {\bf Math 565C - Spring 24 - Homework 4} 

\centerline {\bf Due Wed, March 27}

\bigskip

\no 1. (Oksendal problem 7.3) Let $B_t$ be 1d Brownian motion with $B_0=0$.
Define
$$
X_t = exp(ct + \alpha B_t)
$$
where $c,\alpha$ are constants.
Prove directly from the definition that $X_t$ is a Markov process, i.e.,
that
$$
E[f(X_{t+h}) | \field_t] = E^{X_t}[f(X_h)]
$$
for bounded Borel-measurable $f$. 

\bigskip

\no 2. (based on Oksendal problem 7.8) 

\smallskip
\no (a) Prove that if $\tau_1$ and $\tau_2$ are stopping times, then
$\min \{\tau_1,\tau_2\}$ and $\max \{\tau_1,\tau_2\}$ are stopping times.

\smallskip
\no (b) Let $\tau_n$ be a decreasing sequence of stopping times. Prove that
$\tau = \lim_n \tau_n$ is a stopping time.

\smallskip
\no (c) Let $F$ be closed in $\reals^n$. Let $X_t$ be an Ito diffusion in
$\reals^n$ with continuous paths a.s.
Let $\tau_F$ be the first exit time for $F$, i.e., 
$$
\tau = \inf \{t: X_t \notin F\}
$$
Prove that $\tau$ is a stopping time. Hint: because the paths are continuous
and $F$ is closed,
you can express $\tau$ in terms of $X_t$ at just the rational $t$'s.

\no Optional: Prove the same result with $F$ replaced by an open set.
Hint: An open set can be written as the union of an increasing sequence of
closed sets. 

\bigskip

\no 3. (Oksendal problem 7.10) Let $X_t$ be the geometric Brownian motion
$$
dX_t = r X_t dt + \alpha X_t dB_t
$$
Let $T>0$ be a constant time and $t < T$. 
The goal in this problem is to compute
\be
E^x[X_T|\field_t]
\label{goal}
\ee
in two different ways.


\smallskip
\no (a) Compute it using the Markov property.

\smallskip
\no (b) You can solve this SDE explicitly and then write the solution
as
$$
X_t = x e^{rt} M_t
$$
where
$$
M_t = \exp(\alpha B_t - \frac{1}{2} \alpha^2 t)
$$
We know that $M_t$ is a martingale. (You do not need to show this.)
Use this to compute (\ref{goal}) in a second way. 

\bigskip

\no 4. Consider the SDE 
$$
d X_t = a(t) X_t dt + b(t) X_t \circ dB_t
$$
where the $\circ$ means that we interpret this SDE in the
Stratonovich sense. Solve this SDE with initial condition $X_0$ where
$X_0$ can be random but is independent of the Brownian motion. 
Recall that we solved this SDE when it is interpreted in the Ito sense. 
(This problem should not be hard.) 

\bigskip

\no 5. Suppose that $b(t,x)$ and $\sigma(t,x)$ satisfy the hypotheses of the
existence/(strong) uniqueness theorem. Suppose that $X_t,\bar{X}_t$ are
adapted processes, both of which are continuous in mean square
(recall exercise 3 in homework 2). 
Define
\bea
Y_t = \int_0^t b(s,X_s) ds + \int_0^t \sigma(s,X_s) dB_s \\
\bar{Y}_t = \int_0^t b(s,\bar{X}_s) ds + \int_0^t \sigma(s,\bar{X}_s) d\bar{B_s}
\eea
where $B_t$ and $\bar{B}_t$ are each Brownian motions.

\smallskip

\no (a) 
Prove that the processes $Y_t$ and $\bar{Y}_t$ have the same distribution.

\smallskip
\no (b) Prove that $Y_t$ is continuous in mean square.

\smallskip

\no (c) Now consider the Ito diffusions
\bea
d X_t = b(t,X_t) dt + \sigma(t,X_t) dB_t \\
d \bar{X}_t = b(t,\bar{X}_t) dt + \sigma(t,\bar{X}_t) d\bar{B}_t
\eea
with initial conditions $X_0$ and $\bar{X}_0$ where $X_0$ and $\bar{X}_0$
have the same distribution. Let $X^k_t$ and $\bar{X}^k_t$ be the sequence of
approximations to $X_t$ and $\bar{X_t}$ that come from Picard iteration.
Use (a) and (b) to prove that the processes $X^k_t$ and $\bar{X}^k_t$
have the same distribution for all $k$ and that they are continuous in
mean square for all $k$. This is easy given (a) and (b).

\smallskip

\no (d) Conclude that the processes $X_t$ and $\bar{X}_t$ have the same
distribution, i.e., we have weak uniqueness. Can we conclude that the solutions
$X_t$ and $\bar{X}_t$ are continuous in mean square?

\end{document}
\end
