标题： 多维非线性随机微分方程截断EM方法和对数截断EM方法的最优强收敛率 显示英文标题

标题： The optimal strong convergence rates of the truncated EM and logarithmic truncated EM methods for multi-dimensional nonlinear stochastic differential equations

摘要： 毛学年（2015）提出的截断Euler-Maruyama (EM) 方法被用来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，其收敛率次优。本文的主要目标是展示无无穷小因子的截断EM方法的最优收敛性。此外，对数截断EM方法在多维情形下尚未被研究过，这也是本文的另一个目标。我们将展示保正性的对数截断EM方法求解具有正解的多维SDEs的最优强收敛阶。数值例子被给出以支持我们的理论结论。 显示英文摘要

摘要： 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.