The Numerical Analysis of Stochastic Differential Equations
The Numerical Analysis of Stochastic Differential Equations
复制标题
随机微分方程的数值分析
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
C. Mahony
中科院分区：
文献类型：
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作者：
C. Mahony
本文介绍了随机微分方程数值实验的主要概念和技巧。在第1、2和3节中介绍了随机微分方程、收敛性和随机泰勒展开的基本理论，为了便于理解，在一维情况下介绍了这些理论。第4节和第5节讨论了强近似和弱近似，详细介绍了泰勒型方法以及一些龙格-库塔近似。稳定性和隐式方法出现在第6节，更高维的问题在第7节。数值结果在第8节中讨论，一些软件信息在附录中提供。0引言随着越来越多的现实数学模型被要求考虑到真实的世界系统中的随机效应和影响，随机微分方程（SDEs）对于准确描述此类情况变得至关重要。由于随机微分方程很少有显式解，精确的数值方法是至关重要的，以使其实施可行。由于随机微积分的特点，微分方程的数值分析在某些关键领域与已经发展成熟的常微分方程数值分析领域不同，但这一理论的大部分内容也可以扩展到随机情况。本文将集中在离散时间近似的SDES，这样的计划和问题所产生的实际实施的各种例子的优点和缺点。
This paper provides an introduction to the main concepts and techniques necessary for the someone who wishes to carry out numerical experiments involving stochastic differential equations. The basic theory of SDEs, convergence and stochastic Taylor expansions is presented in sections 1, 2 and 3, and in the one dimensional case for ease of understanding. Strong and Weak approximations are discussed in sections 4 and 5, detailing Taylor-type methods as well as some Runge-Kutta approximations. Stability and implicit methods appear in section 6, and higher dimensional issues are presented in section 7. Numerical results are dealt with in section 8, and some software information is available in the appendix. 0 Introduction As more realistic mathematical models become required to take into account random effects and influences in real world systems stochastic differential equations (SDEs) have become essential in the accurate description of such situations. Since SDEs rarely have explicit solutions, accurate numerical methods are vital in order to make their implementation viable. Due to features of the stochastic calculus the numerical analysis of SDE’s differs in some key areas from the already well-developed area of the numerical analysis of ordinary differential equations, but much of this theory can be extended to the stochastic case also. This paper will concentrate on discrete time approximations of SDEs, the advantages and drawbacks of various examples of such schemes and issues arising from their practical implementation.
影响因子：
2.9
作者：
Higham, DJ
通讯作者：
Higham, DJ
Y.Saito：“随机微分方程数值格式的稳定性分析”SIAMJ.Numer.Anal。
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