研究生签名：______________导师签名：_________________日期：____________
武汉理工大学硕士学位论文
摘要
随机微分方程的理论广泛应用于经济、生物、物理、自动化等领域，然而在 很长一段时间里，由于缺乏有效的求解随机系统的数值方法以及足够强大的计算 机计算能力，在实际问题中，以随机微分方程(组)为代表的描述物理现象的许多 复杂的数学模型或者被束之高阁，或者被迫通过忽略随机因素而简化，均不能得 到很好的应用。可喜的是近十年来，在随机微分方程数值解方面已取得了一些成 就，这意味着由某些随机微分方程描述的数学模型可以借助于计算机进行研究。
First, the background of SDE and the importance of its theoretical solution are introduced. Two of the very important forms of SDE, Ito SDE and Stratonovich SDE, are deduced by stochastic integrals and several lemmas about the moments of stochastic integrals are also given in the paper. In addition, I mention the theorem giving necessary and sufficient conditions for the existence and uniqueness of a solution to SDE and I give representation formulae of solutions of linear SDEs. And the stochastic Taylor series of solution are deduced.
答辩委员会主席
评阅人
2006 年 11 月
独创性声明
本人声明，所呈交的论文是本人在导师指导下进行的研究工作及取得的研究 成果。据我所知，除了文中特别加以标注和致谢的地方外，论文中不包含其它人 已经发表或撰写过的研究成果，也不包含为获得武汉理工大学或其它教育机构的 学位或证书而使用过的材料。与我一同工作的同志对本研究所做的任何贡献均已 在论文中作了明确的说明并表示了谢意。
In the body of the paper, both direct truncation of stochastic Taylor series and a comparison of the Taylor series of the theoretical solution and its corresponding Runge-Kutta form are considered, which lead to Taylor methods and Runge-Kutta methods. For Taylor methods, explicit Euler-Mayaruma method and Milsபைடு நூலகம்ein method are basic for solving Ito SDE(s), on which basis Semi-implicit Euler-Mayaruma method, Semi-implicit Milstein method, implicit Euler-Taylor method and implicit Milstein method are introduced and order 1.5 Taylor method are obtain in the similar way. For Runge-Kutta methods, their application to ordinary differential equation are mentioned at first and the stochastic settings are constructed by comparison. Rooted tree theory simplifies the form of Runge-Kutta methods and two new Runge-Kutta methods of 3 stage explicit (M2) and 3 stage semi-implicit (SIM1) are designed.
For the complexity of stochastic systems, it's very difficult to calculate the representation formulae of solutions of generic SDE. Thus constructing numeric methods is paramount. Nowadays, the research of numerical solution of SDE is still in its nascent state. Convergence and stability need to be considered before developing efficient numerical methods. Stochastic asymptotical stability and that in mean-square sense (MS-stability) of the theoretical solution is introduced in the paper, as well as MS-stability and T-stability.
研究生签名：_____________日期：_________
关于论文使用授权的说明
本人完全了解武汉理工大学有关保留、使用学位论文的规定，即学校有权保 留、送交论文的复印件，允许论文被查阅和借阅；学校可以公布论文的全部或部 分内容，可以采用影印、缩印或其它复制手段保存论文。
(保密的论文在解密后应遵守此规定)
In the end, stability analyses under mean-square sense are performed on concrete methods and numerical simulations are implemented, which illustrate implicit form out-performs semi-implicit, and semi-implicit is better than explicit in stability for every method, and new methods M2, SIM1 have the same relatively higher numerical precision as the classical Runge-Kutta methods (eg. 4 stage explicit (M3) and 2 stage diagonal implicit (DIM1)). Key words: SDE, convergence, stability, Taylor methods, Runge-Kutta methods
I
武汉理工大学硕士学位论文
Abstract
The theory of stochastic differential equation (SDE) was widely applied in the fields of economy, biology, physics and automatization. However, during quite a long period of time, due to the lack of efficient numerical methods for solving stochastic systems and computers with sufficient power, many complicated mathematical models that attempt to represent physical phenomena, such as SDE(s), had been put aside or simplified when applied in practical problems by omitting stochastic factors. Thus these models were just beautiful in form and never fully utilized. Fortunately, in the past decade or so numerical methods for SDE(s) have made some cheering achievements, which predicate some mathematical models represented by SDE(s) are being researched with computers.
and Numerical Simulation
研究生姓名
李炜
指导教师
姓名 黄樟灿 职称 教授 学位 博士 单位名称 理学院 邮编 430070
姓名 副指导教师 单位名称
职称
邮编
申请学位级别 硕士 学科专业名称 应用数学
论文提交日期 2006 年 10 月 论文答辩日期 2006 年 11 月
学位授予单位 武汉理工大学 学位授予日期
本文首先介绍了随机微分方程的背景知识及其理论解的重要性质。其中通过 随机积分导出了 Ito 型和 Stratonovich 型两种重要形式的随机微分方程，并给出 了计算随机积分期望的相关引理；介绍了随机微分方程强解的存在唯一性定理， 对于线性随机微分方程，给出了解的解析表达式；推导了解的随机 Taylor 展开式。