The Direct Richardson pth Order (DRp) Schemes: A New Class of Time Integration Schemes for Stochastic Differential Equations

Abstract

We describe a new family of weak $p$th order accurate SDE time integration schemes, called the direct Richardson $p$th order accurate (DRp) schemes. The DRp schemes use the idea of Richardson extrapolation on Euler time steps, performed by way of an acceptance-rejection algorithm. Previous applications of Richardson extrapolation to the Euler scheme are applicable only when the objective is to estimate a functional of the final distribution of the process. In contrast, provided that the diffusion matrix is strictly positive definite, the DRp class of schemes can be used in all applications which require a weak SDE time integration scheme. Numerical results have been obtained, and a comparison is made between the second- and third-order accurate DRp schemes and other modern SDE time integration schemes, indicating that the DRp schemes incur less error than standard algorithms based on Ito-Taylor expansions, and have similar computational efficiency. Finally, we provide a proof of the convergence properties of the DRp schemes.