Speeding up the Euler scheme for killed diffusions
Let X be a linear diffusion taking values in (โ,r) and consider the standard Euler scheme to compute an approximation to ๐ผ[g(X_T)1_[T<ฮถ]] for a given function g and a deterministic T, where ฮถ=inf{tโฅ 0: X_t โ (โ,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/โ(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.
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