imate solutions contain systematic errors induced by the discretization of the PDEs. Whilst the convergence properties of these methods are usually known as the grid is refined, the magnitude of the error is almost always un- available. This is problematic when performing statistical inference in both the forward and inverse problems, as it is often not possible to construct valid confidence intervals, or Bayesian credible intervals. An alternative approach is to use the Feynman-Kac formulae (FKF) (Kac, 1951), which describe the relationship between the local solutions to certain classes of PDEs and the expectations of initial and boundary functions with respect to the law of complementary stochastic differential equations (SDEs). The FKF approach can only be applied to certain classes of PDEs, but for such cases this approach may have a number of advantages when perform- ing statistical inference. Computationally, the FKF approach may be more efficient in high dimensional problems as the number of grid-points in global solvers increases exponentially with the number of dimensions, whereas the cost of simulating from the complementary SDEs grows sub-exponentially. An additional advantage in the inverse problem is that the number of obser- vations will typically be far fewer than the number of grid points required in a finite differences or finite elements solution, and so obtaining only the required local solutions may further reduce the computational cost. The es- timates of each local solution are also independent, and so can be run in parallel, drastically reducing wall-clock times. Finally, the Monte Carlo er- ror can be studied using, for example, the central limit theorem, offering a 3