文档介绍：摘要随机延迟微分方程广泛应用于工程、物理、医学、,. 第一章简要介绍了随机延迟微分方程的应用背景及其数值分析的研究现状,扼要介绍了本文的主要工作. ,证明了当步长充发小时,显式Euler方法和向后Euler方法都能保持原系统的几乎处处指数稳定性. 第三章构造了求解带Poisson跳的随机延迟微分方程的补偿口方法, Lipschitz条件下,证明方法是1/,,当 i9≤p≤1时,得到了该方法是均方P稳定的结论,这可视为确定性延迟微分方程中P稳定结论的推广. ,证明了当 i9≤目≤1时,对所有的约束网格,方法都是均方稳定的. 第五章研究了非线性随机延迟积分微分方程随机p方法的稳定性. 得到了解析解和数值解均方指数稳定的条件,并获得当丢≤口≤1时, 随机口方法对所有的约束网格都是均方稳定的结论. ,该方法依任意步长保持原系统的均方指数稳定性. 博士学位论文一数值试验结果验证了文中所获理论结论的正确性. 关键词: 随机延迟微分方程,随机延迟积分微分方程,Poisson跳,收敛性,稳定性 ABSTRACT Stochastic delay differentialequations(SDDEs)are widely used in engineering,physics,medical science,biology,economics and iSdi伍cult tofindtheanalytical solutions fortheoverwhelming ma- jority ofSDDEs,therefore,it has important theoretical and practical significance todesign efficientnumerical methods mainly study thestability ofthenumerical methods for SDDEs withPoissonj umps and thestochastic delayintegro-differential equations(SDIDEs).The thesis posed of sixchapters. Inthefirstchapter,the research background oftheSDDEs andthe developments ofthenumerical methods for theSDDEs are introduced. The main contribution ofthisthesisisalsobriefly introduced. Chapter 2focuses on thealmost sure exponential stability ofthe Euler methods forSDDEs with Poisson using thediscrete semimartingale convergence theorem,we proved thatboth theexplicit Eulermethod andthe backward Euler method can reproduce thealmost sure exponential stability oftheunderlying systems provided thatthe stepsize issufficiently small. InChapter stochastic 0methods iS con— structed forthenonlinear SDDEs withPoissonj mean square convergence and stability