tance probability proportional to (5). This requires calculating an upper bound for exp( A ( v )), and upper and lower bounds for φ ( v ) over the current interval. For the Dirichlet problem it is possible to obtain bounds over the entire domain, which can then be applied to every interval. Whether this is a more computationally efficient approach than evaluating tighter bounds in each iteration will be problem dependent. Whilst it is not possible to com- pute the acceptance probability directly as it requires the full sample path, an unbiased acceptance probability can be obtained using Poisson thinning (Lewis and Shedler, 1979), which requires only a finite dimensional repre- sentation of the sample path (Beskos, Papaspiliopoulos and Roberts, 2006a; Chen and Huang, 2013). Accepted simulations from EA can be used to obtain Monte Carlo es- timates of the FKF by using the inverse of the Lamperti transform. Since the function c ( x, t ) is required to be bounded, say by L$_{c}$ and M$_{c}$ , the path weighting term in (3) and (4) can also be estimated using Poisson thinning. A naive Monte Carlo estimate is then given by where each ˆ u ( i ) are of the form 14 ˆ u ( i ) = κ ( η − $^{1}$( Y$_{ˆ}$ $_{t}$)) N$_{p}$ ∏ j =1 ( M$_{c}$ − c ( η − $^{1}$( Y$_{r}$$_{j}$ ) , t − r$_{j}$ ) ) M$_{c}$ − L$_{c}$ · exp { − L$_{c}$ ˆ t } , ˆ u ( x x, t ) = 1 N N ∑ i =1 ˆ u ( i $^{)}$,