The truncated Euler--Maruyama (EM) method, developed by Mao (2015), is used to solve multi-dimensional nonlinear stochastic differential equations (SDEs). However, its convergence rate is suboptimal due to an unnecessary infinitesimal factor. The primary goal of this paper is to demonstrate the optimal convergence of the truncated EM method without infinitesimal factors. Besides, the logarithmic truncated EM method has not been studied in multi-dimensional cases, which is the other goal of this paper. We will show the optimal strong convergence order of the positivity-preserving logarithmic truncated EM method for solving multi-dimensional SDEs with positive solutions. Numerical examples are given to support our theoretical conclusions.

翻译：截断Euler--Maruyama（EM）方法由Mao（2015）提出，用于求解多维非线性随机微分方程（SDEs）。然而，由于存在不必要的无穷小因子，其收敛速率并非最优。本文的主要目标是证明截断EM方法在消除无穷小因子后的最优收敛性。此外，对数截断EM方法在多维情形下的研究尚属空白，这是本文的另一目标。我们将证明该保持正性的对数截断EM方法在求解具有正解的多维SDEs时具有最优强收敛阶。数值算例验证了理论结论。