Speeding up the Euler scheme for killed diffusions

Abstract

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