Speeding up the Euler scheme for killed diffusions

Let X be a linear diusion taking values in (`; r) and consider the standard Euler scheme to compute an approximation to E[g(XT )1[T<]] for a given function g and a deterministic T, where = infft 0 : Xt =2 (`; r)g. It is well-known since Gobet [21] that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1= p 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 [9]. 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