随机微分方程数值解稳定性研究综述

Investigation on stability of numerical schemes of stochastic differential equations: A survey

本文回顾了近年来随机微分方程数值方法的稳定性的研究成果.作为相关话题,收敛性问题也有所涉猎.以经典It8型随机微分方程、中立型随机泛函微分方程、Markov跳随机微分方程和Poisson跳随机微分方程为代表,主要介绍了几类数值方法稳定性研究的成果.这些方法包括常见的Euler-Maruyama方法、Backward Euler-Maruyama方法、θ方法、分步方法等.文中分析了关于稳定性等价性定理经典论文的学术思路,提出了随机微分方程数值计算与仿真所面临的挑战及所要解决的问题.

In this paper,a survey is given for the investigation on the stability of numerical schemes of stochastic differential equations in the past years.As a related topic ,the convergence of the schemes is involved.The paper introduces the achieved results by literatures for the classical Ito^ stochastic differential equations, stochastic functional differential equations of the neutral type, and the stochastic differential equations with Markov or Poisson jumps.The involved numerical schemes include the Euler-Maruyama scheme, the Backward Euler-Maruyama scheme, the 0 scheme ,and the split-step scheme, etc.The paper analyzes the academic thoughts in some classical literatures on the stability equivalence theorems and proposes some problems and challenges for further investigations on the numerical computations and simulations of stochastic differential equations at the end of the paper.