Speeding up the Euler scheme for killed diffusions

Let \(X\) be a linear diffusion taking values in \((\ell ,r)\) and consider the standard Euler scheme to compute an approximation to \(\mathbb{E}[g(X_{T}){\mathbf{1}}_{\{T<\zeta \}}]\) for a given function \(g\) and a deterministic \(T\), where \(\zeta =\inf \{t\geq 0: X_{t} \notin (\ell ,r)\}\). 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 \(1/\sqrt{N}\) 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.