【作者】 高海音；

【导师】 王克；

【作者基本信息】 东北师范大学 ， 应用数学， 2006， 博士

【摘要】 本文主要研究了随机微分方程解的存在性和有界性理论,首先将随机微分方程和随机泛函微分方程解的存在唯一性的充分条件进行了相应的改进。接下来,系统给出了随机微分方程解的各种有界性定义,利用Lyapunov直接方法,建立了一系列随机微分方程解的有界性定理,并给出应用举例,表明了所获理论结果的可实现性,最后引入随机微分方程解的增长阶估计的定义,将随机微分方程解的有界性问题统一为随机微分方程解的增长阶估计问题,并建立了一系列随机微分方程解的增长阶估计定理,同时给出了应用举例。 全文共分成四章: 第一章作为准备知识给出了本文要用到的相关内容,其中包括随机微分方程的主要基本概念和论文中要用到的基本定理及重要不等式; 第二章介绍了现有随机微分方程及随机泛函微分方程解的存在唯一性定理,并将随机微分方程及随机泛函微分方程解的存在唯一性定理进行了相应的改进; 第三章给出随机微分方程解的各种有界性定义,利用Lyapunov直接方法,建立了一系列随机微分方程解的有界性定理,并给出应用举例; 第四章给出随机微分方程解的增长阶估计的定义及随机微分方程解的增长阶估计定理,将随机微分方程解的有界性问题统一为随机微分方程解的增长阶估计问题,并给出应用举例。 在论文的最后,总结了论文的创新点,并提出了论文的改进方向以及研究中所参考的主要文献。更多还原

【Abstract】 Existence and boundedness theory for the stochastic differential equations are mainly investigated in this thesis, which improve the existence-uniqueness theorem for the stochastic differential equations and stochastic functional differential equations. A series of new definitions have been established. A number of new criteria on the boundedness for the solutions of stochastic differential equations are obtained by using Lyapunov straight method. The new results will be illustrated by many examples. New definitions and theorem have been established on the increase order estimate for the solutions of stochastic differential equations.The whole thesis is divided into five chapters.Chapter 1. It offers some relative knowledge including basic concepts, theorem and inequality for the stochastic differential equations.Chapter 2. It offers the results of existence-uniqueness theorem for the stochastic differential equations obtained by [15]. Then follows we do further improvement to existence-uniqueness theorem for the stochastic differential equations and stochastic differential functional equations.Chapter 3. It offers new definitions on the boundedness for the solutions of stochastic differential equations as well as new theorem on the boundedness for the solutions of stochastic differential equations.Chapter 4. It offers new definitions and theorem on the estimate increase order for the solutions of stochastic differential equations. Which are illustrated by examples.At the end of the thesis,The innovations, further study direction and many related references are listed.更多还原