time approximations of the SDE, such as the Euler-Maruyama approximation (Kloeden and Platen, 1992), for example. This approach was used in Her- bei and Berliner (2014) to perform inference on an ocean-circulation model. However, such numerical schemes lead to biased estimates of the FKF due to discretization errors. In this article we demonstrate how advancements in two areas can be used to obtain unbiased estimates of the FKF for a wider range of mod- els. The first is the recent development of exact algorithms (EA). These are rejection algorithms that use Brownian motion proposals to obtain finite representations of SDEs of interest, and are exact in the sense that they do not contain discretization errors. In particular we extend the localisation approach of Chen and Huang (2013) to multiple dimensions, and design an adaptive scheme to simulate the exact first passage time from a hyperrect- angle, allowing us to directly obtain Monte Carlo estimates of solutions to certain Dirichlet problems. The EA approach imposes restrictions on the drift and volatility functions of the complementary SDE, and so is not al- ways available. The second is the development of debiasing techniques for expectations with respect to SDEs. Debiasing techniques construct unbiased estimates using a weighted sum of biased estimates that are obtained from numerical approximations. Since debiasing only requires that we are able to simulate from the SDE using approximate (but convergent) schemes, it has fewer restrictions than EA, and so can be more generally applied, albeit at the cost of larger work-variance products. 5