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%  Homework for the course "Stochastic Calculus", 
%  Fall semester, 2002, Jonathan Goodman.


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{\scriptsize Stochastic Calculus, Fall 2002 
(http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)} \hfill 

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Assignment 9, last assignment.   
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\noindent
Given  December 5, due December 19 
Last revised December 5.
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{\bf Focus:} Multidimensional diffusions and PDE
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\begin{description}

\item[1.] The Ornstein Uhlenbeck process is described by the pair of SDE's:
\begin{equation}
dV_t = -\gamma V_t dt + \alpha dW_t \; ,
\label{V}  \end{equation}
\begin{equation}
dX_t = V_t dt \; .
\label{X}   \end{equation}
As originally proposed by Einstein and refined by Ornstein and Uhlenbeck,
this is a model for the motion of a small particle in a fluid.  Equation (1) 
describes the speed of the particle, $V_t$.  It says that there is a random 
forcing, $\alpha dW_t$, caused by collisions from molecules of the fluid, and 
a drag, $-\gamma V_t$, caused by friction with the fluid.  Equation (2)
just says that the rate of change of the position is given by the speed.
\begin{description}
\item[a.] The equation (1) may be solved using the method of integrating
factors.  Show that if we multiply both sides by the integrating factor
$e^{\gamma t}$, we get
$$
d \bigl( e^{\gamma t}V_t\bigr) = \alpha e^{\gamma t}dW_t \; ,
$$
in the Ito sense.
\item[b.] Use this to derive a formula
$$
V_T = e^{-\gamma T}V_0 + \int_0^T m(T,t)dW_t \; ,
$$
with a specific formula for $m$.
\item[c.] Use this to show that $V_t \sim {\cal N}(\mu_t,\sigma_t^2)$, and
find formulae for $\mu_t$ and $\sigma_t$.  
\item[d.] Suppose $u(v,t)$ is the PDF for $V_t$, write the forward equation
PDE satisfied by $u$.  This may be expressed in the form 
$\partial_t u = {\cal L}^* u$, where ${\cal L}$ is the generator of the 
process.  (By convention, the generator is the operator that goes in the
backward equation, so it's adjoint is what goes in the forward equation.)
\item[e.] From the answer to part c, together with the formula for the
density of a ${\cal N}(\mu,\sigma^2)$ random variable, find a formula for 
$u(v,t)$.
\item[f.] Verify by explicit differentiation that this formula satisfies the
forward equation from part d.
\item[g.] Find the limits
$$
\mu_0 = \lim_{t \to \infty}\mu_t\;\;,\;\;\;\;\mbox{and} \;\;\;\;
\sigma_0 = \lim_{t \to \infty}\sigma_t  \; .
$$
Show that $u(v)$, the ${\cal N}(\mu_0,\sigma_0^2)$ density, satisfies
${\cal L}^*u = 0$.  This is the steady state velocity distribution.  The
relationship between $\alpha$ and $\gamma$ for a given $\sigma_0$ is a
special case of what physicists call the fluctuation dissipation theorem.
\item[h.] Express $X_T$ as an integral involving $W_t$ for $t \leq T$.
\item[i.] Use the integral expressions for $X_t$ and $V_t$ to show that 
$(X_t,V_t)$ is a two dimensional normal and calculate
the mean and variance-covariance matrix.
\item[j.] Use this to write a formula for $u(x,v,t)$, the joint PDF for
$(X_t,V_t)$.  Notice that this is a smooth function of $x$ and $v$ for $t>0$
even though (1)(2) represents a degenerate diffusion.  Not all degenerate diffusions 
have this property.
\item[k.] Write the forward equation PDE for $u(x,v,t)$.
\item[l.] Verify that the formula from part j satisfies this PDE.
\item[m.] Show that for large $t$ the variance of $X_t$ has the approximate
form $\mbox{var}(X_t) \approx D^2 t$.  Find a formula for $D$ in terms of
$\gamma$ and $\alpha$.  This formula, and its physical interpretation,
was a significant part fo the research that won Einstein the Nobel Prize.
\end{description}

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