  The Feynman-Kac formulae (FKF) express local solutions of partial
differential equations (PDEs) as expectations with respect to some
complementary stochastic differential equation (SDE). Repeatedly sampling paths
from the complementary SDE enables the construction of Monte Carlo estimates of
local solutions, which are more naturally suited to statistical inference than
the numerical approximations obtained via finite difference and finite element
methods. Until recently, simulating from the complementary SDE would have
required the use of a discrete-time approximation, leading to biased estimates.
In this paper we utilize recent developments in two areas to demonstrate that
it is now possible to obtain unbiased solutions for a wide range of PDE models
via the FKF. The first is the development of algorithms that simulate diffusion
paths exactly (without discretization error), and so make it possible to obtain
Monte Carlo estimates of the FKF directly. The second is the development of
debiasing methods for SDEs, enabling the construction of unbiased estimates
from a sequence of biased estimates.
