Abstract
Let X be a linear diffusion taking values in (l,r) and consider the standard Euler scheme to compute an approximation to E[g(XT)1[T<ζ]] for a given function g and a deterministic T, where ζ=inf{t ≥ 0: XT ∉ (l,r)}. It is well known since Gobet (Weak approximation of killed diffusion using Euler schemes. Stochastic Processes and their Applications, 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 discretisatons. 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 (Diffusion transformations, Black–Scholes equation and optimal stopping. 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.