随机常微分方程的二步方法及其数值分析

发布时间：2018-09-01 13:00

【摘要】：随机常微分方程已经广泛应用于金融系统、数量经济、控制系统、系统生物等研究领域.由于随机系统本身的复杂性,一般情况下很难得到方程解析解的显式表达式.因此,对随机常微分方程的数值方法进行研究就显得十分必要.本论文主要研究随机常微分方程的数值方法,提出分裂步二步Maruyama方法、全隐式二步Maruyama方法和全隐式二步Milstein方法,并分别分析相应数值方法的均方相容性、均方收敛性与均方线性稳定性.另外,提出数值求解带泊松跳的随机常微分方程的二步Maruyama方法,分析该算法的均方相容性、均方收敛性与均方线性稳定性.第一章,介绍随机常微分方程的基本理论,简单回顾随机常微分方程数值解法的发展历史与研究现状,并说明本文的主要研究内容和结果.第二章,简要介绍概率论中的一些基础知识,随机过程和随机积分的基本概念,以及Ito公式和Ito-Taylor展式的相关结论.第三章,提出数值求解随机常微分方程的分裂步二步Maruyama方法,分析方法的均方相容性、均方收敛性及均方线性稳定性,给出分裂步二步Adarms-Bashforth Maruya-ma方法和分裂步二步Adarms-Moulton Maruyama方法的均方线性稳定性区域,并通过数值算例验证算法的均方收敛性和均方稳定性的理论结果.第四章,提出数值求解随机常微分方程的全隐式二步Maruyama方法,分析其均方相容性、均方收敛性及均方线性稳定性,给出全隐式二步Adams-Bashforth Maruyama方法和全隐式二步Adams-Moulton Maruyama方法的均方线性稳定性区域.最后,通过数值算例验证该算法的均方收敛性和均方稳定性结果.第五章,提出数值求解随机常微分方程的全隐式二步Milstein方法,对该方法的均方相容性、均方收敛性及均方线性稳定性进行分析,给出全隐式二步Adams-Bashforth Milstein方法和全隐式二步Adams-Moulton Milstein方法的均方线性稳定性区域.数值例子表明理论结果的正确性.第六章,提出数值求解带泊松跳的随机常微分方程的二步Maruyama方法,分析该方法的均方相容性、均方收敛性及均方线性稳定性,研究二步Adams-Bashforth Maruya-ma 和二步 Adams-Moulton Maruyama 方法的均方线性稳定性区域.最后,通过数值例子验证该算法的均方收敛性和均方稳定性的理论结果.
[Abstract]:Stochastic ordinary differential equations have been widely used in the fields of financial system, quantity economy, control system, system biology and so on. Because of the complexity of the stochastic system itself, it is difficult to obtain the explicit expression of the analytic solution of the equation in general. Therefore, it is necessary to study the numerical method of stochastic ordinary differential equation. In this paper, the numerical methods of stochastic ordinary differential equations are studied. The split step two-step Maruyama method, the fully implicit two-step Maruyama method and the fully implicit two-step Milstein method are proposed, and the mean square compatibility of the corresponding numerical methods is analyzed respectively. Mean square convergence and mean square linear stability. In addition, a two-step Maruyama method for solving stochastic ordinary differential equations with Poisson hopping is presented. The mean square compatibility, mean square convergence and mean square linear stability of the algorithm are analyzed. In the first chapter, the basic theory of stochastic ordinary differential equation is introduced, and the development history and research status of numerical solution of stochastic ordinary differential equation are briefly reviewed, and the main contents and results of this paper are explained. In the second chapter, we briefly introduce some basic knowledge of probability theory, the basic concepts of stochastic process and stochastic integral, and the relevant conclusions of Ito formula and Ito-Taylor expansion. In chapter 3, a split step two-step Maruyama method for solving stochastic ordinary differential equations is presented. The mean square compatibility, mean square convergence and mean square linear stability of the method are analyzed. The mean square linear stability regions of split step two step Adarms-Bashforth Maruya-ma method and split step two step Adarms-Moulton Maruyama method are given, and the theoretical results of mean square convergence and mean square stability of the algorithm are verified by numerical examples. In chapter 4, a fully implicit two-step Maruyama method for solving stochastic ordinary differential equations is presented. The mean square compatibility, mean square convergence and mean square linear stability are analyzed. The mean square linear stability regions of the fully implicit two-step Adams-Bashforth Maruyama method and the fully implicit two-step Adams-Moulton Maruyama method are given. Finally, the mean square convergence and mean square stability of the algorithm are verified by numerical examples. In chapter 5, a fully implicit two-step Milstein method for solving stochastic ordinary differential equations is presented. The mean square compatibility, mean square convergence and mean square linear stability of the method are analyzed. The mean square linear stability regions of the fully implicit two-step Adams-Bashforth Milstein method and the fully implicit two-step Adams-Moulton Milstein method are given. Numerical examples show that the theoretical results are correct. In chapter 6, a two-step Maruyama method for solving stochastic ordinary differential equations with Poisson hopping is presented. The mean square compatibility, mean square convergence and mean square linear stability of the method are analyzed. The mean square linear stability region of two step Adams-Bashforth Maruya-ma and two step Adams-Moulton Maruyama methods is studied. Finally, a numerical example is given to verify the theoretical results of the mean square convergence and mean square stability of the algorithm.
【学位授予单位】：上海师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.81

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