已知随机Euler格式收敛于具有全局Lipschitz连续漂移和扩散系数的随机微分方程的精确解。最近的结果将这种收敛性推广到至多线性增长的系数。对于超线性增长的系数，根据[Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 1041-1063]，有限时间内的强均方收敛性仍是一个开放问题。在本文中，我们对这个问题给出了否定回答，并证明对于一大类具有非全局Lipschitz连续系数的随机微分方程，Euler近似在有限时间点既不在强均方意义下收敛，也不在数值弱意义下收敛于精确解。更糟的是，在有限时间点，精确解与数值近似之差在强均方意义和数值弱意义下发散至无穷。
英文摘要：
The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz continuous drift and diffusion coefficient. Recent results extend this convergence to coefficients which grow at most linearly. For superlinearly growing coefficients finite-time convergence in the strong mean square sense remained an open question according to [Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 1041-1063]. In this article we answer this question to the negative and prove for a large class of stochastic differential equations with non-globally Lipschitz continuous coefficients that Euler's approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense.
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities
On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients
Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme
Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients
University of Strathclyde（斯特拉斯克莱德大学）；University of Oxford（牛津大学）
Stability of the semi-tamed and tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition
Center for Research in Computational and Applied Mechanics (CERECAM)（计算与应用力学研究中心）；University of Cape Town（开普敦大学）；Technische Universität Chemnitz（开姆尼茨工业大学）
Convergence of the tamed Euler scheme for stochastic differential equations with Piecewise Continuous Arguments under non-Lipschitz continuous coefficients
Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）
Strong convergence rates of a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise
Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）；School of Statistics and Data Science, Jiangxi University of Finance and Economics（江西财经大学统计与数据科学学院）
Strong convergence of the backward Euler approximation for the finite element discretization of semilinear parabolic SPDEs with non-global Lipschitz drift driven by additive noise
Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization
Archimedes, Athena Research Center（阿基米德雅典研究中心）；National Technical University of Athens（雅典国立技术大学）
Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction
Sichuan University（四川大学）；National Key Laboratory of Fundamental Algorithms and Models for Engineering Simulation, Sichuan University（工程仿真基础理论及应用国家重点实验室，四川大学）
Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods
