Speeding up the Euler scheme for killed diffusions

Let X be a linear diffusion taking values in (`, 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 / ∈ (`, r)}. It is well-known since [16] 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 [6]. 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.

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