Speeding up the Euler scheme for killed diffusions

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