两类随机微分方程的均方渐近概周期解

发布时间：2018-03-17 23:06

本文选题：随机微分方程　切入点：均方渐近概周期解　出处：《哈尔滨理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：随机微分方程是在解决某些具有随机现象的问题而建立起来的一类方程。随机微分方程在诸多领域有着广泛的应用，而随机微分方程的均方概周期类型解的存在性和唯一性在随机过程理论和概周期型函数理论的基础上更加具有研究意义。本文主要讨论了两类随机微分方程均方渐近概周期解的存在性和唯一性。全文内容如下：第一部分介绍了目前概周期型函数理论和随机微分方程理论的背景知识和主要研究成果，以及今后的发展趋势。第二部分研究了一类非线性随机微分方程的均方渐近概周期解的存在唯一性。首先给出有关概周期随机过程的部分理论知识，介绍了一类一致渐近稳定的C0半群的有关知识，然后讨论了联合连续函数的渐近概周期性质，利用该性质、Fubini定理、Holder不等式以及Banach不动点原理讨论了该方程均方渐近概周期解的存在唯一性。第三部分在第二部分的理论基础上，讨论了一类非自治随机微分方程的均方渐近概周期解的存在唯一性。首先给出解决该类方程所需的理论知识和相关内容，然后介绍了一类算子开方族的基本概念，，再应用Fubini定理、Holder不等式以及Banach不动点原理讨论了该类方程均方渐近概周期解的存在唯一性。
[Abstract]:Stochastic differential equation is a kind of equation which is established in solving some problems with random phenomena. Stochastic differential equation is widely used in many fields. However, the existence and uniqueness of mean square almost periodic type solutions of stochastic differential equations are of great significance on the basis of stochastic process theory and almost periodic function theory. In this paper, we mainly discuss two kinds of stochastic differential equations. Existence and uniqueness of asymptotically almost periodic solutions. The text reads as follows:. The first part introduces the background knowledge and main research results of almost periodic function theory and stochastic differential equation theory, as well as the development trend in the future. In the second part, we study the existence and uniqueness of mean-square asymptotically almost periodic solutions for a class of nonlinear stochastic differential equations. Firstly, we give some theoretical knowledge about almost periodic stochastic processes, and introduce some knowledge about a class of uniformly asymptotically stable C _ 0 Semigroups. Then the asymptotically almost periodic property of joint continuous function is discussed and the existence and uniqueness of the mean square asymptotic almost periodic solution of the equation are discussed by using the Banach fixed point principle and the Fubini theorem. In the third part, on the basis of the theory of the second part, we discuss the existence and uniqueness of the mean square asymptotic almost periodic solution for a class of nonautonomous stochastic differential equations. Then we introduce the basic concept of a class of open square family of operators, and discuss the existence and uniqueness of mean square asymptotically almost periodic solutions of this class of equations by using Fubini theorem and Banach fixed point principle.
【学位授予单位】：哈尔滨理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O211.63

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