Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients II: finite moments and higher-order schemes

TL;DR

This paper investigates the weak convergence of explicit numerical schemes for SDEs with super-linear coefficients, focusing on finite moments and higher-order schemes, and provides a systematic approach to establish convergence orders.

Contribution

It introduces a systematic method to analyze weak convergence orders of explicit schemes for SDEs with limited moments, extending previous work to higher-order schemes.

Findings

Numerical schemes achieve predictable weak convergence orders.

Limited moments influence the convergence behavior of schemes.

Numerical examples confirm theoretical convergence orders.

Abstract

This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a blowup of moments of solutions and numerical solutions and thus affect the convergence of numerical methods. Wang et al. (2023, IMA J. Numer. Anal.) have analyzed weak convergence of one-step numerical schemes when solutions to SDEs have all finite moments. Therein some modified Euler schemes have been discussed about their weak convergence orders. In this work, we explore the effects of limited orders of moments on the weak convergence of a family of explicit schemes. The schemes are based on approximations/modifications of terms in the Ito-Talyor expansion. We provide a systematic but simple way to establish weak convergence orders for these schemes. We present several numerical examples of these schemes and show their weak convergence orders.