Speeding up the Euler scheme for killed diffusions

Let X X be a linear diffusion taking values in ( ,r) ( ℓ , r ) and consider the standard Euler scheme to compute an approximation to E[g(XT)1_<ζ \] E [ g ( X T ) 1 { T < ζ } ] for a given function g g and a deterministic T T , where ζ =inf ≥ 0: Xₜ ∉ ( ,r)\ 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 1 / N with N N being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to $1/N$ 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.