Suppose that we have a
particular type of parabolic partial differential
equation and we know what that the solution
at some fixed time T is u[x, T] = ?[x]. For
a particular (x,t), the Feynman-Kac formula
says that u(x, t), is the expected value of
? [XT-t] conditioned upon the fact that Xo =
x. We can determine the stochastic process Xt
using information encoded in the PDE. Rather
than numerically calculate the path integral
associated to this expected value, we instead
consider calculating u(x,t) as the integral
of u(y,T) = ?(y) against pT-t [x,y], where pT-t
[x,y] is the transition density of the process
Xt . |

We will discuss two general
scenarios. First, we will consider the scenario
when pT-t [x,y] is easy to compute and the numerical
solution can therefore be produced directly
by way of a integration procedure in a program
like Mathematica. Second, we will consider some
cases when pT-t[x,y] must be approximated. In
these cases, we introduce a recursive procedure
to approximate u[x,t]. Finally, we will illustrate
this procedure using examples from mathematical
finance. |