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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 8.
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\noindent
Given November 11, due November 18. Last revised, November 11.\\
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{\bf Objective:} Diffusions and diffusion equations.
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% The questions
\begin{description}
\item[1.] An Ornstein Uhlenbeck process is a stochastic process that
satisfies the stochastic differential equation
\begin{equation}
dX(t) = - \gamma X(t) dt + \sigma dW(t) \; .
\label{OU} \end{equation}
\begin{description}
\item[a.] Write the backward equation for $f(x,t) = E_{x,t}[V(X(T)]$.
\item[b.] Show that the backward equation has (Gaussian) solutions of the form
$f(x,t) = A(t)\exp(-s(t) (x-\xi(t))^2/2)$. Find the differential equations
for $A$, $\xi$, and $s$ that make this work.
\item[c.] Show that $f(x,t)$ does not represent a probability distribution,
possibly by showing that $\int_{-\infty}^{\infty}f(x,t)dt$ is not a constant.
\item[d.] What is the large time behavior of $A(t)$ and $s(t)$?
What does this say about the nature of an Ornstein Uhlenbeck reward that
is paid long in the future as a function of starting position?
\end{description}
\item[2.]The forward equation:
\begin{description}
\item[a.] Write the forward equation for $u(x,t)$ which is the probability
density for $X(t)$.
\item[b.] Show that the forward equation has Gaussian solutions of the form
$$
u(x,t) = \frac{1}{\sqrt{2\pi\sigma(t)^2}}e^{-(x-\mu(t))^2/2\sigma^2(t) } \; .
$$
Find the appropriate differential equations for $\mu$ and $\sigma$.
\item[c.] Use the explicit solution formula for (\ref{OU}) from assignment 7
to calculate $\mu(t) = E[X(t)]$ and $\sigma(t) = \mbox{var}[X(t)]$.
These should satisfy the equations you wrote for part b.
\item[d.] Use the approximation from (\ref{OU}):
$\Delta X \approx -\gamma X \Delta t + \sigma \Delta W$ (and the
independent increments property) to express $\Delta \mu$ and
$\Delta (\sigma^2)$ in terms of $\mu$ and $\sigma$ and get yet another
derivation of the answer in part b. Use the definitions of $\mu$
and $\sigma$ from part c.
\item[e.] Differentiate $\int_{-\infty}^{\infty} x u(x,t) dx$ with
respect to $t$ using the forward equation to find a formula for
$d\mu/dt$. Find the formula for $d\sigma/dt$ in a similar way from
the forward equation.
\item[f.] Give an abstract argument that $X(t)$ should be a Gaussian
random variable for each $t$ (something is a linear function of something),
so that knowing $\mu(t)$ and $\sigma(t)$ determines $u(x,t)$.
\item[g.] Find the solutions corresponding to $\sigma(0) = 0$ and
$\mu(0) = y$ and use them to get a formula for the transition probability
density (Green's function) $G(y,x,t)$. This is the probability density
for $X(t)$ given that $X(0) = y$.
\item[h.] The transition density for Brownian motion is
$G_B(y,x,t) = \frac{1}{\sqrt{2\pi t}}\exp(-(x-y)^2/2t)$.
Derive the transition density for the Ornstein Uhlenbeck process
from this using the Cameron Martin Girsanov formula (warning: I have not
been able to do this yet, but it must be easy since there is a simple
formula for the answer. Check the bboard.).
\item[i.] Find the large time behavior of $\mu(t)$ and $\sigma(t)$.
What does this say about the distribution of $X(t)$ for large $t$ as a
function of the starting point?
\end{description}
\item[3.] Duality:
\begin{description}
\item[a.] Show that the Green's function from part 2 satisfies the backward
equation as a function of $y$ and $t$.
\item[b.] Suppose the initial density is $u(x,0) = \delta(x-y)$ and that
the reward is $V(x) = \delta(x-z)$. Use your expressions for the
corresponding forward solution $u(x,t)$ and backward solution $f(x,t)$
to show by explicit integration that
$\int_{-\infty}^{\infty}u(x,t)f(x,t) dx$ is independent of $t$.
\end{description}
\end{description}
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