more natural approach for statistical inference than the error bounds of finite difference and finite element methods. The unbiased local solutions can then be embedded in either a Bayesian or maximum likelihood framework in order to study the inverse problem. 

Analytical solutions to the FKF are rarely available, but it is usually possible to obtain Monte Carlo estimates by sampling a finite number of random paths from the complementary SDE. Owing to the difficulty of sim- ulating from SDEs, existing applications of the FKF have suffered various limitations. The random walk on spheres (WOS) algorithm (Muller, 1956), for example, was proposed to obtain Monte Carlo solutions to the Dirichlet problem of Laplace’s equation. The WOS algorithm has since been extended (see for example (Hwang and Mascagni, 2001; Hwang, Mascagni and Given, 2003)), but is still restricted to a narrow class of models, in particular re- quiring that the drift and diffusion functions of the complementary SDE are constant. In all cases solutions are approximate as simulated paths are ter- minated in a region near the boundary, rather than on the boundary itself. A more recent modification is the random walk on rectangles (WOR) al- gorithm (Deaconu and Lejay, 2006). If the domain is a polytope then the WOR algorithm can be used to simulate the absorption time exactly, and so the only error in the estimate of the FKF is that associated with the Monte Carlo approximation. In many situations the drift and/or volatility functions will be non-constant, preventing the application of the WOS and WOR algo- rithms. In such cases estimates of the FKF can be obtained by using discrete 

