随机微分方程的截断θ方法的收敛性分析

发布时间：2018-03-11 07:31

本文选题：随机微分方程　切入点：单调性条件　出处：《广西师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：一般情况下,在研究随机微分方程数值方法的收敛性时,需要方程的漂移项与扩散项同时满足全局Lipschitz条件和线性增长条件.然而由于线性增长条件太强,现实生活中的绝大多数SDEs模型并不满足此条件,因此本文在局部Lipschitz条件及单调性条件下为随机微分方程构造了一种新的半隐式数值方法,即截断θ方法,并建立了相关的收敛性理论.本文结构如下:第1章为绪论.主要介绍随机微分方程的相关背景,研究现状,本文的创新点和主要内容.第2章为预备知识.主要介绍本文的相关基础知识和本文所用符号的含义.第3章构造了半隐式的截断θ方法,并在局部Lipschitz条件,单调性条件及扩散项的多项式条件下,证明了截断θ方法的两种连续类型的数值解是强收敛的,最后用数值实验验证了本章的理论结果.第4章讨论了构造的截断θ方法在给定条件下的收敛速度,并且证明了该算法q阶矩的收敛阶近似于1/2,且用数值实验验证了本章的结论.最后对本文做了总结和展望.
[Abstract]:In general, in studying the convergence of numerical methods for stochastic differential equations, it is necessary to satisfy both the global Lipschitz condition and the linear growth condition for both the drift term and the diffusion term of the equation. However, the linear growth condition is too strong. Most SDEs models in real life do not satisfy this condition. In this paper, a new semi-implicit numerical method, truncated 胃 method, is constructed for stochastic differential equations under local Lipschitz condition and monotonicity condition. The structure of this paper is as follows: chapter 1 is the introduction. The innovation and main contents of this paper. Chapter 2 is the preparatory knowledge. It mainly introduces the basic knowledge of this paper and the meaning of the symbols used in this paper. In chapter 3, the semi-implicit truncation 胃 method is constructed, and the local Lipschitz condition is obtained. Under the monotonicity condition and the polynomial condition of diffusion term, it is proved that the numerical solutions of two continuous types of truncation 胃 method are strongly convergent. Finally, numerical experiments are used to verify the theoretical results of this chapter. Chapter 4 discusses the convergence rate of the constructed truncation 胃 method under given conditions. It is proved that the convergence order of the qth-order moment of the algorithm is approximately 1 / 2, and the conclusion of this chapter is verified by numerical experiments. Finally, the conclusion of this paper is summarized and prospected.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63

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