\documentclass[12pt]{article}
\usepackage{graphicx}% Include figure files
\usepackage{amssymb}
\usepackage{amsfonts}
%\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}

\newcommand{\ba}[1]{\begin{array}{#1}}
\newcommand{\ea}{\end{array}}
\newcommand{\arr}[2]{\ba{#1} #2 \ea}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand {\no}  {\noindent}
\newcommand{\borel}{\mathcal{B}}
\newcommand{\f}{\mathcal{F}}
\newcommand{\e}{\mathcal{E}}
\newcommand{\ra}{\rightarrow}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\field}{\mathcal{F}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\V}{\mathcal{V}}

\begin{document}

\centerline {\bf Math 565C - Spring 24 - Homework 3} 
% wed sept 5
\centerline {\bf Due Fri, March 1}

\bigskip

\no 1. (Evans problem 41) Let $X_t$ solve the Langevin equation with the
initial condition that $X_0$ is normal with mean $0$ and variance
$\frac{\sigma^2}{2b}$ and independent of the Brownian motion. We saw in class
that $X_t$ is a Gaussian process with mean zero. Find the covariance of
this process. The Langevin equation is discussed in Evans pp. 86-88.

\bigskip

\no 2. (Oksendal 5.7) The mean-reverting Orstein Uhlenbeck process is given by
$$
dX_t = (m-X_t)dt + \sigma dB_t
$$
where $m,\sigma$ are constants.

\smallskip
\no (a) Solve this equation

\smallskip
\no (b) Find $E[X_t]$ and $var(X_t)$. For more fun you can find the covariance
of the process, but this is optional. 

\bigskip

\no 3. (Evans problem 43) One approach to trying to solve an SDE is to
assume the solution has the form $X_t = g(t,B_t)$. (This assumption is not
valid in general, but for some SDE's it works.)

\smallskip
\no (a) Use this method to solve
\bea
dX_t &=& -\frac{1}{2} e^{-2X_t} \, dt + e^{-X_t} \, dB_t, \\
X(0) &=& x_0
\eea
where $x_0$ is a constant. 

\smallskip
\no (b) Show that the solution blows up at a finite random time. Explain
(very briefly) why this does not contradict our existence theorem which
give existence of a solution for all time. 


\newpage

\no 4. (Oksendal 4.13) Let $X_t$ be an Ito process of the form
$$
d X_t = u(t,\omega) dt + dB_t 
$$
This is a martingale only if $u=0$. When $u$ is nonzero we can get a martingale
by multiplying $X_t$ by a suitable exponential. Assume that $u$ is bounded
and define $Y_t = X_t M_t$ where
$$
M_t = \exp\left(-\int_0^t u(s,\omega) dB_s
-\frac{1}{2} \int_0^t u^2(s,\omega) \, ds \right)
$$
Use Ito's formula to prove that $Y_t$ is a martingale with respect to the
Brownian filtration. See Oksendal for a comment on how this is related to
the Girsanov theorem. 

\bigskip

\no 5. (Oksendal 5.14) Complex Brownian motion $B(t)$ is defined as
$$
B(t) = B_1(t) + i B_2(t)
$$
where $B_1,B_2$ are independent (real-valued) Brownian motions. 

\smallskip

\no (a) Let $F(z)$ be an analytic function.  Define $Z_t = F(B(t))$.
Show that $ dZ_t = F^\prime(B(t)) \, dB(t)$. (Here $F^\prime(z)$ is the
usual complex derivative of $F(z)$.) $Z_t$ is a complex valued process.
You can compute $dZ_t$ by applying Ito's lemma to its real and imaginary
parts.
Hint: Cauchy-Riemann equations. 

\smallskip

\no (b) Solve the SDE
$$
dZ_t = \alpha Z_t \, dB(t) 
$$
where $\alpha$ is a constant. 

\end{document}
\end



