Speeding up the Euler scheme for killed diffusions

Let X be a linear diffusion taking values in (ℓ,r) and consider the standard Euler scheme to compute an approximation to E[g(X T)1 {T<ζ}] for a given function g and a deterministic T, where ζ=inf{t≥0:X t∉(ℓ,r)}. It is well known since Gobet (Stoch. Process. Appl. 87:167–197, 2000) that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1/N with N being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N, i.e., the optimal rate in the absence of killing, using the theory of recurrent transformations developed in Çetin (Ann. Appl. Probab. 28:3102–3151, 2018). Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.