An extended Milstein scheme for effective weak approximation of diffusions

TL;DR

This paper introduces an extended Milstein scheme for discretizing multi-dimensional diffusion processes, achieving a weak convergence rate of one with fewer error terms, leading to more accurate estimations especially when diffusion coefficients are small.

Contribution

The paper presents a new explicit Milstein-type scheme that reduces leading-order error terms and improves accuracy over classical schemes like EM and Milstein for weak approximation.

Findings

The new scheme has a weak convergence rate of one.

It outperforms EM and Milstein schemes in simulations with small diffusion parameters.

No performance difference observed between EM and Milstein schemes in experiments.

Abstract

We propose a straightforward and effective method for discretizing multi-dimensional diffusion processes as an extension of Milstein scheme. The new scheme is explicitly given and can be simulated using Gaussian variates, requiring the same number of random variables as Euler-Maruyama (EM) scheme. We show that the proposed scheme has a weak convergence rate of one, which is consistent with other classical schemes like EM/Milstein schemes but involves fewer leading-order error terms. Due to the reduction of the error terms, the proposed scheme is expected to provide a more accurate estimation than alternative first-order schemes. We demonstrate that the weak error of the new scheme is effectively reduced compared with EM/Milstein schemes when the diffusion coefficients involve a small parameter. We conduct simulation studies on Asian option pricing in finance to showcase that our proposed scheme significantly outperforms EM/Milstein schemes, while interestingly, we find no differences in the performance between EM and Milstein schemes.