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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 5.
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\noindent
Given October 1, due October 21. Last revised, October 7.\\
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{\bf Objective:} Brownian Motion.
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% The questions
\begin{description}
\item[1.] Suppose $h(x)$ has $h^{\prime}(x) > 0$ for all $x$ so that
there is at most one $x$ for each $y$ so that $y=h(x)$. Consider the
process $Y_t = h(X_t)$, where $X_t$ is standard Brownian motion.
Suppose the function $h(x)$ is smooth. The answers to the questions
below depend at least on second derivatives of $h$.
\begin{description}
\item[a.] With the notation $\Delta Y_t = Y_{t+\Delta t} - Y_t$, for
a positive $\Delta t$, calculate $a(y)$ and $b(y)$ so that
$E[\Delta Y_t \mid {\cal F}_t] = a(Y_t)\Delta t + O(\Delta t^2)$ and
$E[\Delta Y_t^2 \mid {\cal F}_t] = b(Y_t)\Delta t + O(\Delta t^2)$.
\item[b.] With the notation $f(Y_t,t) = E[V(Y_T) \mid {\cal F}_t]$,
find the backward equation satisfied by $f$. (Assume $T>t$.)
\item[c.] Writing $u(y,t)$ for the probability density of $Y_t$,
{\em use the duality argument} to find the forward equation satisfied by $u$.
\item[d.] Write the forward and backward equations for the special case
$Y_t = e^{cX_t}$. Note (for those who know) the similarity of the backward
equation to the Black Scholes partial differential equation.
\end{description}
\item[2.] Use a calculation similar to the one we used in class to show
that $Y_T = X_T^4 - 6 \int_0^T X_t^2 dt$ is a martingale. Here $X_t$
is Brownian motion.
\item[3.] Show that $Y_t = \cos(kX_t)e^{k^2 t/2}$ is a martingale.
\begin{description}
\item[a.] Verify this directly by first calculating (as in problem 1) that
$$
E[Y_{t+\Delta t} \mid {\cal F}_t] = Y_t + O(\Delta t^2)\; .
$$
Then explain why this implies that $Y_t$ is a martingale exactly
(Hint: To show that $E[Y_{t^{\prime}} \mid {\cal F}_t] = Y_t$, divide
the time interval $(t,t^{\prime})$ into $n$ small pieces and let $n \to \infty$.
\item[b.]
Verify that $Y_t$ is a martingale using the fact that a certain function
satisfies the backward equation. Note that, for any function $V(x)$,
$Z_t = E[V(X_T)\mid {\cal F}_t]$ is a martingale (the tower property).
Functions like this $Z$ satisfy backward equations.
\item[c.] Find a simple intuition that allows a supposed martingale to grow
exponentially in time.
\end{description}
\item[4.] Let $A_{x_0,t}$ be the event that a standard Brownian motion
starting at $x_0$ has $X_{t^{\prime}} > 0$ for all $t^{\prime}$ between
$0$ and $t$. Here are two ways to verify the large time asymptotic
approximation $P(A_{x_0,t}) \approx \frac{1}{\sqrt{2\pi}} \frac{2x_0}{\sqrt{t}}$.
\begin{description}
\item[a.] Use the formula from ``Images and reflections'' to get
\begin{eqnarray*}
P(A_{x_0,t}) & = & \int_0^{\infty} u(x,t) dx \\
& \approx & \frac{1}{\sqrt{2\pi t}} \int_0^{\infty} e^{-x^2/2t}
\left( e^{xx_0/t} - e^{-xx_0/t}\right) dx \; .
\end{eqnarray*}
The change of variables $y=x/\sqrt{t}$ should make it clear how to
approximate the last integral for large $t$.
\item[b.] Use the same formula to get
\begin{equation}
\frac{-d}{dt} P(A_{x_0,t}) = \frac{1}{\sqrt{2\pi}} \frac{2x_0}{t^{3/2}}
e^{-x_0^2/2t} \; .
\end{equation}
Once we know that $ P(A_{x_0,t}) \to 0$ as $t \to \infty$, we can
estimate its value by integrating (1) from $t$ to $\infty$
using the approximation $e^{\mbox{\scriptsize \em const}/t} \approx 1$
for large $t$.
Note: There are other hitting problems for which $P(A_t)$ does not go to
zero as $t \to \infty$. This method would not work for them.
\end{description}
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