Speeding up the Euler scheme for killed diffusions

06/28/2021

∙

by   Umut Çetin, et al.

∙

0

∙

share

Let X be a linear diffusion taking values in (ℓ,r) and consider the standard Euler scheme to compute an approximation to 𝔼[g(X_T)1_[T<ζ]] for a given function g and a deterministic T, where ζ=inf{t≥ 0: X_t ∉ (ℓ,r)}. It is well-known since <cit.> 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 <cit.>. 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.