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A Computational Approach to Path Integral Solutions of Parabolic PDEs

David "Jeff" Hamrick
Center for Computational Science and Mathematics & Statistics
Boston University
October 29, 2004
 
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.
 
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