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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
\begin{center}
\large
Assignment 6.
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\noindent
Given October 21, due October 28. Last revised, October 21.\\
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{\bf Objective:} Forward and Backward equations for Brownian motion.
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% The questions
The terms forward and backward equation refer to the equations the
probability density of $X_t$ and $E_{x,t}[V(X_T)]$ respectively.
The integrals below are easily done if you use identities such as
$$
\frac{1}{\sqrt{2\pi\sigma^2}}
\int_{-\infty}^{\infty} x^{2n} e^{-x^2/2\sigma^2}dx
= \sigma^{2n} \cdot (2n-1)(2n-3)\cdots 3 \; .
$$
You should not have to do any actual integration for these problems.
\begin{description}
\item[1.] Solve the forward equation with initial data
$$
u_0(x) = \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2} \; .
$$
\begin{description}
\item[a.] Assume the solution has the form
\begin{equation}
u(x,t) = \left( A(t) x^2 + B(t) \right) g(x,t) \; \; ,\;\;\;\;
g(x,t) = G(0,x,t+1) = \frac{1}{\sqrt{2\pi(t+1)}} e^{-x^2/2(t+1)} \; .
\end{equation}
Then find and solve the ordinary differential equations for $A(t)$ and $B(t)$
that make (1) a solution of the forward equation.
\item[b.] Compute the integrals
$$
u(x,t) = \int_{-\infty}^{\infty} u_0(y) G(y,x,t) dy \; .
$$
This should give the same answer as part a.
\item[c.] Sketch the probability density at time $t=0$, for small time, and
for large time. Rescale the large time plot so that it doesn't look flat.
\item[d.] Why does the structure seen in $u(x,t)$ for small time (the double
hump) disappear for large $t$?
\item[e.] Show in a rough way that a similar phenomenon happens for any
initial data of
the form $u_0(x) = p(x) g(x,0)$, where $p(x)$ is an even nonnegative polynomial.
When $t$ is large, $u(x,t)$ looks like a simple Gaussian, no matter what $p$
was.
\end{description}
\item[2.] Solve the backward equation with final data $V(x) = x^4$.
\begin{description}
\item[a.] Write the solution in the form
\begin{equation}
f(x,t) = x^4 + a(t) x^2 + b(t) \; .
\end{equation}
Then find and solve the differential equations that $a(t)$ and $b(t)$ must
satisfy so that (2) is the solution of the backward equation.
\item[b.] Compute the integrals
$$
f(x,t) = \int_{-\infty}^{\infty} G(x,y,T-t) V(y) dy \; .
$$
This should be the same as your answer to part a.
\item[c.] Give a simple explanation for the form of the formula for
$f(0,t) = b(t)$ in terms of moments of a Gaussian random variable.
\end{description}
\item[3.] Check that
$$
\int_{-\infty}^{\infty} u(x,t) f(x,t) dx
$$
is independent of $t$.
\end{description}
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