  We consider hypoelliptic Kolmogorov equations in $n+1$ spatial dimensions,
with $n\geq 1$, where the differential operator in the first $n$ spatial
variables featuring in the equation is second-order elliptic, and with respect
to the $(n+1)$st spatial variable the equation contains a pure transport term
only and is therefore first-order hyperbolic. If the two differential
operators, in the first $n$ and in the $(n+1)$st co-ordinate directions, do not
commute, we benefit from hypoelliptic regularization in time, and the solution
for $t>0$ is smooth even for a Dirac initial datum prescribed at $t=0$. We
study specifically the case where the coefficients depend only on the first $n$
variables. In that case, a Fourier transform in the last variable and standard
central finite difference approximation in the other variables can be applied
for the numerical solution. We prove second-order convergence in the spatial
mesh size for the model hypoelliptic equation $\frac{\partial u}{\partial t} +
x \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2}$ subject to
the initial condition $u(x,y,0) = \delta (x) \delta (y)$, with $(x,y) \in
\mathbb{R} \times\mathbb{R}$ and $t>0$, proposed by Kolmogorov, and for an
extension with $n=2$. We also demonstrate exponential convergence of an
approximation of the inverse Fourier transform based on the trapezium rule.
Lastly, we apply the method to a PDE arising in mathematical finance, which
models the distribution of the hedging error under a mis-specified derivative
pricing model.
