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 di- rectly. The second is the development of debiasing methods for SDEs, enabling the construction of unbiased estimates from a sequence of biased estimates. Keywords: Debiasing; Exact algorithms; Feynman Kac formula; Multi- level Monte Carlo; Partial differential equations; Stochastic differential equa- tions. Partial differential equations (PDEs) describe many natural and physical phenomena, and have long been the foundation of mathematical models used for the study of complex behaviour. These can be considered forward models, in the sense that for some given set of model parameters they generate a set of possible observations. In many situations the goal is to use observations to estimate the model parameters, and in so doing, better understand the underlying processes of the system. This is known as the inverse problem, which is detailed in, for example, Stuart (2010). In most cases analytical solutions to PDEs of interest are unavailable, and so solutions are usually obtained using a numerical approximation. Approaches such as finite element and finite difference methods provide discrete space-time approximations to the solutions at specified grid-points over regions of interest. These approx- 2 1. Introduction