A new extrapolation method for weak approximation schemes with applications

TL;DR

This paper introduces a universal extrapolation method to enhance weak approximation schemes for stochastic differential equations, building upon Fujiwara's sixth order scheme and extending it to higher orders.

Contribution

It presents a novel extrapolation technique that generalizes Fujiwara's scheme, enabling the construction of higher-order weak approximation schemes from lower-order ones.

Findings

Developed a general method for higher-order weak schemes

Extended Fujiwara's sixth order scheme to order 2m

Provided a framework for extrapolating from Ninomiya-Victoir scheme

Abstract

We review Fujiwara's scheme, a sixth order weak approximation scheme for the numerical approximation of SDEs, and embed it into a general method to construct weak approximation schemes of order $ 2m $ for $ m \in \mathbf{N} $. Those schemes cannot be seen as cubature schemes, but rather as universal ways how to extrapolate from a lower order weak approximation scheme, namely the Ninomiya-Victoir scheme, for higher orders.