具有非利普希茨系数的随机微分方程解的存在性与唯一性

发布时间：2018-04-16 05:40

本文选题：非利普希茨 + 强解　；参考：《中国科学技术大学》2017年硕士论文

【摘要】：随机微分方程在统计物理、控制论、种群遗传学、电路理论、金融期权等诸多领域均有重要应用。研究一个随机微分方程的首要课题是其解的存在唯一性问题(包括解是否爆炸的问题)。人们对随机微分方程定义了多种类型的解,又对它的解定义了多种意义下的唯一性。有利普希茨系数项的随机微分方程解的存在唯一性问题均已经有了明确的结论。而利普希茨连续属于较强的条件,故在实际问题中,需要考虑的方程的系数并不总是符合此条件。故有非利普希茨系数项的方程一直是随机分析领域中研究的热点。本文主要研究的问题就是有非利普希茨系数项的随机微分方程解的存在唯一性问题(包括解是否爆炸的问题)。本文结构如下:第一章是本文的绪论部分。第二章简要地介绍布朗运动的随机积分与It6公式,以便于后面引出随机微分方程这一概念。第三章我们给出了本文所针对的随机微分方程的形式,并严格定义了其弱解、强解以及解的各种唯一性,还有解的爆炸时,并且还说明了一下解的各种存在性与唯一性的内在联系。第四章,我们主要分类总结了现有的关于非利普希茨系数项方程解的存在唯一性的重要结论。第五章,我们主要介绍了近年来的一些关于非利普希茨系数项的方程解的存在唯一性的重要结论。在第六章,我们首先回顾了在第四、五章中综述过的三个重要结论,对它们稍作了一点改进推广,并证明了它们;而后,我们基于前人的研究思想与成果,提出了两个关于非利普希茨系数项的方程解的唯一性的新的定理,并证明了它们,以此解决了两种条件下的非时齐类型的方程解的唯一性问题。
[Abstract]:Stochastic differential equations have important applications in the fields of statistical physics, cybernetics, population genetics, circuit theory, financial options and so on.The most important problem in the study of a stochastic differential equation is the existence and uniqueness of its solution (including the problem of whether the solution is exploded or not).Many types of solutions are defined for stochastic differential equations, and uniqueness is defined for their solutions.The existence and uniqueness of solutions for stochastic differential equations with Lipschitz coefficients have been clearly concluded.Lipschitz continuity is a strong condition, so the coefficients of equations that need to be considered in practical problems do not always meet this condition.Therefore, the equation with non-Lipschitz coefficient term has always been a hot topic in the field of stochastic analysis.The main problem of this paper is the existence and uniqueness of the solution of stochastic differential equations with non-Lipschitz coefficients (including the problem of whether the solution is exploded or not).The structure of this paper is as follows: the first chapter is the introduction of this paper.In chapter 2, the stochastic integral and It6 formula of Brownian motion are introduced briefly, so as to introduce the concept of stochastic differential equation.In chapter 3, we give the form of stochastic differential equation, and define strictly its weak solution, strong solution and all kinds of uniqueness of solution, and the explosion of solution.The existence and uniqueness of the solution are also explained.In chapter 4, we classify and summarize the existing important conclusions on the existence and uniqueness of the solutions of the non-Lipschitz coefficient equation.In chapter 5, we mainly introduce some important conclusions about the existence and uniqueness of the solution of the non-Lipschitz coefficient equation in recent years.In the sixth chapter, we first review the three important conclusions summarized in chapters 4 and 5, improve and generalize them a little bit, and prove them, and then, based on the previous research ideas and results,In this paper, two new theorems on the uniqueness of solutions of equations with non-Lipschitz coefficients are presented, and they are proved to solve the uniqueness problem of solutions of non-homogeneous equations under two conditions.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63

【参考文献】