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\begin{document}

\centerline {\bf Math 565C - Spring 24 - Homework 5} 

\centerline {\bf Due Wed, April 10}

\bigskip

\no 1. (Oksendal problem 7.3) Let $B_t$ be 1d Brownian motion with $B_0=0$.
Define
$$
X_t = x \, exp(ct + \alpha B_t)
$$
where $c,\alpha$ are constants and $x$ is non-random.
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. Let $X_t$ be the one-dimensional diffusion
$$
dX_t = b \, dt + d B_t
$$
with $b \neq 0$. This is just Brownian motion with a drift:
$X_t= bt +B_t$. Fix $a>0$. Start the process at $X_0 = x_0$ with $0 < x_0 < a$. 
Let $\tau$ be the time when the process exits $(0,a)$, i.e.,
$$
\tau = \inf \{ t>0:X_t \notin (0,a)\}
$$

\no (a) Prove that $E[\tau]<\infty$

\no (b) Compute $P(X_\tau = a)$ and $E[\tau]$.

\smallskip

\no  Hint: In class we did this for $b=0$ by using the Dynkin formula.

\medskip

\no 3. Oksendal 7.9
% hitting for geometric BM

\medskip

\no 4. Oksendal 8.2
% show solution of a certain PDE is given by .... involving geometric Brownian motion. 

\medskip

\no 5. Oksendal 8.3
% Kolmogorov forward and FP equation

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