We consider 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 their numerical solutions. When solutions to SDEs have all finite moments, weak convergence of numerical schemes has been investigated in [Wang et al (2023), Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients, IMA Journal numerical analysis]. Some modified Euler schemes have been analyzed for weak convergence. In this work, we present a family of explicit schemes of first and second-order weak convergence based on classical schemes for SDEs. We explore the effects of limited moments on these schemes. We provide a systematic but simple way to establish weak convergence orders for schemes based on approximations/modifications of drift and diffusion coefficients. We present several numerical examples of these schemes and show their weak convergence orders.

翻译：本文研究在单侧Lipschitz条件下求解随机微分方程(SDE)的一步格式的弱收敛问题。已知超线性系数可能导致精确解及其数值解矩的爆破。当SDE解的所有矩均有限时，数值格式的弱收敛性已在[Wang等人(2023)，具有超线性系数SDE强逼近格式的弱误差分析，IMA数值分析期刊]中得到研究。已有若干修正欧拉格式的弱收敛性被分析。本工作基于经典SDE格式，提出了一类具有一阶和二阶弱收敛性的显式格式族。我们探讨了有限矩对这些格式的影响，提供了一种系统而简洁的方法来建立基于漂移项与扩散项近似/修正格式的弱收敛阶。我们给出了这些格式的若干数值算例，并展示了其弱收敛阶。