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% Homework for the course "Stochastic Calculus",
% Fall semester, 2004, Jonathan Goodman.
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{\scriptsize Stochastic Calculus, Fall 2004
(http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2004/)} \hfill
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\large
Assignment 7.
\normalsize
\end{center}
\noindent
Given November 4, due November 11. Last revised, November 9.\\
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{\bf Objective:} Pure and applied mathematics.
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The first problems are strictly theoretical. They illustrate how clever some
rigorous proofs are. The inequality (3) serves the following function:
We want to understand something about the entire path $F_k$ for
$0 \leq k \leq n$. We can get bounds on $F_k$ for particular values of
$k$ by calculating expectations (e.g. $E[F_k^2]$). Then (3) uses this to
say something about the whole path. As an application, we will
have an easy proof of the convergence of the approximations to the Ito
integral for all $t\leq T$ once we can prove it at the single time $T$.
\begin{description}
\item[1.] Let $F_k$ be a discrete time nonnegative martingale.
Let $M_n = \max_{0\leq k \leq n} F_k$ be its {\em maximal function}.
This problem is the proof that
\begin{equation}
P(M_n > f ) \leq \frac{1}{f} E[F_n {\bf 1}_{M_n \geq f}] \; .
\end{equation}
The proof also shows that if $F_k$ is
any martingale and $M_n = \max_{0\leq k \leq n} \left|F_k\right|$ its
maximal function, then
\begin{equation}
P(M_n > f ) \leq \frac{1}{f} E[\left|F_n\right| {\bf 1}_{M_n \geq f}] \; .
\end{equation}
These inequalities are relatives of {\em Markov's inequality} (also called
{\em Chebychev's inequality}, though that term us usually applied to an
interesting special case), which says that
if $X$ is a nonnegative random variable then
$P(X>a) < \frac{1}{a}E[X {\bf 1}_{X>a}]$, or, if $X$ is any random variable,
that $P(|X|>a) < \frac{1}{a}E[|X| {\bf 1}_{|X|>a}]$.
\begin{description}
\item[a.] Let $A$ be the event $A = \left\{M_n \geq f \right\}$.
Write $A$ as a disjoint union of disjoint events $B_k \in {\cal F}_k$
so that $F_k(\omega) \geq f$ when $\omega \in B_k$.
Hint: If $M_n \geq f$, there is a first $k$ with $F_k \geq f$.
\item[b.] Since $F_k(\omega) \geq f$ for $\omega \in B_k$, show that
$P(B_k) \leq \frac{1}{f} E[{\bf 1}_{B_k}F_k]$ (the main step in the
Markov/Chebychev inequality)).
\item[c.] Use the martingale property and the tower property to show that
${\bf 1}_{B_k}F_k = E[{\bf 1}_{B_k}F_n \mid {\cal F}_k]$ so
$E[{\bf 1}_{B_k}F_k] = E[{\bf 1}_{B_k}F_n]$.
Do this for discrete probability if that helps you.
\item[d.] Add these to get (1).
\item[e.] We say $F_k$ is a {\em submartingale} if
$G_k \leq E[G_n \mid {\cal F}_k]$ (warning: submartingales go up, not down).
Show that if $F_k$ is any martingale, then $|F_k|$ is a submartingale.
Show (1) applies to nonnegative submartingales so (2) applies to general
martingales, positive or not.
\end{description}
\item[2.] Let $M$ be any nonnegative random variable. Define
$\mu(f) = P(M \geq f)$, which is related to the CDF of $M$.
Use the definition of the abstract integral to show that
$E[M] = \int_0^{\infty} \mu(f) df$ and
$E[M^2] = 2 \int_0^{\infty} f \mu(f) df$.
These formulas work even if the common value is infinite.
If $G$ is another nonnegative random variable, show that
$E[GM] = \int_0^{\infty} E[G{\bf 1}_{M\geq f}] df$.
Of course, one way to do this is to formulate a single general formula
that each of these is a special case of.
\item[3.] Use the formulas of part 2 together with Doob's inequality (2)
to show that
$$
E[M_n^2] \leq 2 E[M_n \left|F_n\right|] \; ,
$$
so
\begin{equation}
E[M_n^2] \leq 2 E\left[F_n^2 \right] \; .
\end{equation}
(It will help to use the {\em Cauchy Schwarz inequality}
$E[XY] \leq (E[X^2]E[Y^2])^{1/2}$.)
\end{description}
Now some more concrete examples. We can think of martingales as absolutely
non mean reverting. The inequality (3) expresses that fact in one way:
the maximum of a martingale is comparable to its value at the final
time, on average. The Ornstein Uhlenbeck process is the simplest continuous
time mean reverting process, a continuous time anologue of the simple
urn model.
\begin{description}
\item[4.]An {\em Ornstein Uhlenbeck} process is an adapted process $X(t)$
that satisfies the Ito differential equation
\begin{equation}
dX(t) = -\gamma X(t) dt + \sigma dW(t) \; .
\end{equation}
We cannot use Ito's lemma to calculate $dX(t)$ because $X(t)$ is not a
function of $W(t)$ and $t$ alone.
\begin{description}
\item[a.] Examine the definition of the Ito integral and verify that if
$g(t)$ is a differentiable function of $t$, and $dX(t) = a(t) dt + b(t) dW(t)$,
with a random but bounded $b(t)$, then
$d(g(t) X(t)) = \dot{g}(t) X(t) dt + g(t) dX(t)$.
It may be helpful to use the Ito isometry formula (paragraph 1.17 of lecture
7).
\item[b.] Bring the drift term to the left side of (4), multiply by
$e^{\gamma t}$ and integrate (using part a) to get
$$
X(T) = e^{-\gamma T}X(0) + \sigma \int_0^T e^{-\gamma (T-t)}dW(t) \; .
$$
\item[c.] Conclude that $X(t)$ is Gaussian for any $T$ (if $X(0)$ is)
and that the probability density for $X(T)$ has a limit as $T \to \infty$.
Find the limit by computing the mean and variance.
\item[d.]Contrast the large time behavior of the Ornstein Uhlembeck process
with that of Brownian motion.
\end{description}
\item[5.] Show that $e^{ikW(t)+k^2t/2}$ is a martingale using Ito
differentiation.
\end{description}
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