\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 2} 
% wed sept 5
\centerline {\bf Due ???}  

\smallskip

\no 1. Let $B_t$ be Brownian motion. Let $\sigma$ be a constant.
For $t \ge 0$ let
$$
M_t = \exp(\sigma B_t - \frac{1}{2} \sigma^2 t)
$$
In class I showed (or will show) that $M_t$ is a martingale using the Ito
formula. Prove that $M_t$ is a martingale just using the definition of
a martingale and properties of Brownian motion. 

% Check if some function of BM is a martingale in spirit of 3.4,3.5,3.6

\medskip

\no 2. (Exercise 3.10 in Oksendal) Let $f(t,\omega) \in \V(0,T)$.
Suppose that there exist constants $K<\infty$ and $\epsilon>0$ such
that
$$
E[|f(s,\omega)-f(t,\omega)|^2] \le K |t-s|^{1+\epsilon}, \quad s,t \in [0,T]
$$
This is a form of ``smoothness'' of $f$ as a function of $t$. 

Let $\{ t^{(n)}_j\}$ be a sequence of partitions such that the largest gap
in the $n$th partition goes to zero as $n \ra \infty$. 
For each partition let $\overline{t}^{(n)}_j \in [t^{(n)}_j,t^{(n)}_{j+1})$
Prove that
$$
\sum_j f(\overline{t}^{(n)}_j,\omega) (B_{t_{j+1}}-B_{t_j})
$$
converges to the Ito integral $\int_0^T f(t,\omega) \, dB_t$ as
$n \ra \infty$. 
The Stratonovich integral is defined by taking $\overline{t}^{(n)}_j$
to be the midpoint $(t^{(n)}_j + t^{(n)}_{j+1})/2$. So this result shows that 
for such an $f$ the Ito integral is equal to the Stratonovich integral. 

\medskip

\no 3. (Exercise 3.13 in Oksendal)

\smallskip

\no (a) A stochastic process $X_t$ is continuous in mean square if
for all $t$ we have $E[X_t^2]<\infty$ and 
$$
\lim_{s \ra t} E[(X_s-X_t)^2] =0 
$$
For example, Brownian motion is continuous in mean square.

Let $f : \reals \ra \reals$ be a Lipschitz function. This means
that there is a constant $C$ such that
$$|f(x)-f(y)| \le C |x-y|, \quad \forall x,y 
$$
Prove that if $X_t$ is continuous in mean square then $f(X_t)$ is
continuous in mean square.  (This is easy.) In particular this shows that
any Lipschitz function of Brownian motion is continuous in mean square. 

\smallskip

\no (b) Let $X_t$ be a stochastic process which is continuous in mean
square. Fix $T < \infty$. 
Suppose that $X_t(\omega) \in \V(0,T)$ so that the Ito integral of
$X_t(\omega)$ is defined. Prove that
$$
\int_0^T X_t \, dB_t = \lim_{n \ra \infty} \int_0^T \phi_n(t,\omega) \, dB_t
$$
where
$$
\phi_n(t,\omega) = \sum_j X_{t_j}(\omega) 1_{[t_j^{(n)},t_{j+1}^{(n)})}(t)
$$
and $\{ t^{(n)}_j\}$ is a sequence of partitions such that the largest gap
in the $n$th partition goes to zero as $n \ra \infty$. 

\medskip

\no 4.
Define $h_n(t,x)$ by 
$$
h_n(x,t) = \frac{(-t)^n}{n!} \, \exp({\frac{x^2}{2t}}) \, \frac{d^n}{dx^n} \,
\exp(-\frac{x^2}{2t}) 
$$
(These functions are related to the Hermite polynomials.)
Prove that
$$
d h_{n+1}(B_t,t) = h_n(B_t,t) dB_t
$$
In particular this shows that $h_{n+1}(B_t,t)$ is a martingale. 

More examples of the use of the Ito lemma can be found in exercises 
4.1,4.2,4.3 (product rule), 4.11 in Oksendal, all of which have
solutions in the book. 

\medskip

\no 5. (Oksendal 4.13)
Let $X_t$ be the Ito process given by $d X_t = u(t,\omega) dt + dB_t$.
Assume that $u$ is bounded. $X_t$ is not a martingale unless $u=0$.
We can modify $X_t$ to create a martingale as follows. Define
$$
Y_t = X_t \exp\left(- \int_0^t u(s,\omega) \, dB_s
- \frac{1}{2} \int_0^t  u^2(s,\omega) \, ds \right)
$$
Use Ito's formula to prove that $Y_t$ is a martingale with respect to the
$\sigma$-field generated by $B_t$. 

\end{document}
\end



