This paper introduces balanced implicit two-step Maruyama methods for solving Itô stochastic differential equations. Such methods, compared to those corresponding standard linear two-step Maruyama methods, have better mean-square properties, which is confirmed by a comparison of the stability regions for some particular two-step Maruyama methods. Moreover, the convergence order is investigated which proves the convergence of the presented methods with the order 12 in the mean-square sense. Numerical results are reported to show the convergence properties and the stability properties of the balanced implicit two-step Maruyama methods. The stability analysis and numerical results show that the proposed methods are very promising methods for stiff stochastic differential equations.

中文翻译：

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用于随机微分方程的平衡隐式两步 Maruyama 方法


本文介绍了求解 Itô 随机微分方程的平衡隐式两步 Maruyama 方法。与相应的标准线性两步 Maruyama 方法相比，这些方法具有更好的均方特性，这可以通过比较某些特定两步 Maruyama 方法的稳定性区域来证实。此外，研究了收敛阶数，证明了所提出的方法在均方意义上与 12 阶的收敛性。数值结果表明了平衡隐式两步 Maruyama 方法的收敛特性和稳定性特性。稳定性分析和数值结果表明，所提方法是非常有前途的刚性随机微分方程方法。