G-布朗运动驱动的随机微分方程平均原理研究
发布时间：2018-11-28 07:54
【摘要】：G-布朗运动是在次线性期望(或者说非线性期望)理论框架下构造出来的一类新型的布朗运动。G-布朗运动及对应的G-随机分析理论是分析金融市场中风险测度及更广泛的不确定决策理论的基本工具。因此,G-期望在金融世界中得到越来越广泛的应用。G-布朗运动的引入很好的弥补了以往布朗运动研究中的不足。在至今形成的分析方法中,随机平均法是一个较为常用并且很重要的近似分析方法。随机平均原理作为一类近似解析方法的理论基础,为研究复杂微分方程的性质提供了一种方便而又简单的途径。次线性期望框架下的G-布朗运动驱动的随机微分方程的随机平均原理还鲜有人研究。同时,考虑到全局或局部Lipschitz条件在实际的模型中是相当苛刻的,因此研究非Lipschitz条件下(比Lipschitz条件更弱的条件)G-布朗运动作用下随机微分方程的随机平均原理具有重要的理论和实际意义。论文主要工作如下:1、我们首先考虑Lipschitz条件下G-布朗运动作用的随机微分方程的随机平均原理。在适合的条件下,证明了经过近似处理的方程的解与原始方程的解在均方意义下和依容度意义下的收敛性。平均原理作为平均法的理论基础给了我们简化方程复杂性的理论依据。通过我们的近似处理,我们可以忽略原始复杂的系统,只考虑平均后的系统,因为它们在均方意义下是等价的。2、考虑到全局或局部Lipschitz条件在实际的模型中是相当苛刻的,因此我们研究了非Lipschitz条件下(比Lipschitz条件更弱的条件)G-布朗运动作用的随机微分方程的随机平均原理具有重要的理论和实际意义。证明了经过近似处理的方程的解与原始方程的解在均方意义下和依容度意义下的收敛性。
[Abstract]:G- Brownian motion is a new type of Brownian motion constructed under the framework of sublinear expectation (or nonlinear expectation) theory. G- Brownian motion and corresponding G- stochastic analysis theory is an analysis of stroke in financial market. Risk measures and the broader theory of uncertain decision-making basic tools. Therefore, G- expectation is more and more widely used in the world of finance. The introduction of G- Brownian motion makes up for the shortcomings of previous researches on Brownian motion. Among the analytical methods developed so far, the stochastic averaging method is a relatively common and important approximate analysis method. As the theoretical basis of a class of approximate analytical methods, the stochastic averaging principle provides a convenient and simple way to study the properties of complex differential equations. The stochastic averaging principle of G- Brownian motion driven stochastic differential equations under the framework of sublinear expectation is seldom studied. At the same time, considering that the global or local Lipschitz conditions are quite harsh in the actual model, Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations under non-Lipschitz conditions (weaker than Lipschitz conditions) under the action of G-Brownian motion. The main work of this paper is as follows: 1. We first consider the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under Lipschitz condition. Under suitable conditions, the convergence of the solution of the approximate equation and the solution of the original equation in the sense of mean square and tolerance is proved. As the theoretical basis of the averaging method, the averaging principle gives us a theoretical basis for simplifying the complexity of the equation. Through our approximation, we can ignore the original complex systems and only consider the average systems, because they are equivalent in the mean square sense. 2, considering that the global or local Lipschitz conditions are quite harsh in the actual model. Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under non-Lipschitz conditions (weaker than the Lipschitz condition). It is proved that the solution of the approximate equation and the solution of the original equation are convergent in the sense of mean square and tolerance.
【学位授予单位】：西北农林科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
本文编号：2362272
[Abstract]:G- Brownian motion is a new type of Brownian motion constructed under the framework of sublinear expectation (or nonlinear expectation) theory. G- Brownian motion and corresponding G- stochastic analysis theory is an analysis of stroke in financial market. Risk measures and the broader theory of uncertain decision-making basic tools. Therefore, G- expectation is more and more widely used in the world of finance. The introduction of G- Brownian motion makes up for the shortcomings of previous researches on Brownian motion. Among the analytical methods developed so far, the stochastic averaging method is a relatively common and important approximate analysis method. As the theoretical basis of a class of approximate analytical methods, the stochastic averaging principle provides a convenient and simple way to study the properties of complex differential equations. The stochastic averaging principle of G- Brownian motion driven stochastic differential equations under the framework of sublinear expectation is seldom studied. At the same time, considering that the global or local Lipschitz conditions are quite harsh in the actual model, Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations under non-Lipschitz conditions (weaker than Lipschitz conditions) under the action of G-Brownian motion. The main work of this paper is as follows: 1. We first consider the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under Lipschitz condition. Under suitable conditions, the convergence of the solution of the approximate equation and the solution of the original equation in the sense of mean square and tolerance is proved. As the theoretical basis of the averaging method, the averaging principle gives us a theoretical basis for simplifying the complexity of the equation. Through our approximation, we can ignore the original complex systems and only consider the average systems, because they are equivalent in the mean square sense. 2, considering that the global or local Lipschitz conditions are quite harsh in the actual model. Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under non-Lipschitz conditions (weaker than the Lipschitz condition). It is proved that the solution of the approximate equation and the solution of the original equation are convergent in the sense of mean square and tolerance.
【学位授予单位】：西北农林科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
【参考文献】
相关期刊论文 前3条
1 Xue-peng BAI;Yi-qing LIN;;On the Existence and Uniqueness of Solutions to Stochastic Differential Equations Driven by G-Brownian Motion with Integral-Lipschitz Coefficients[J];Acta Mathematicae Applicatae Sinica(English Series);2014年03期
2 Yong Xu;Rong Guo;Wei Xu;;A limit theorem for the solutions of slow fast systems with fractional Brownian motion[J];Theoretical & Applied Mechanics Letters;2014年01期
3 ;On Representation Theorem of G-Expectations and Paths of G-Brownian Motion[J];Acta Mathematicae Applicatae Sinica(English Series);2009年03期
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