G-期望框架下的几类随机微分方程的研究
发布时间：2022-11-04 21:37
在本文中,我们讨论G－期望框架下由G-布朗运动和G-Levy过程驱动的几类随机微分方程.论文由五个部分组成,结构如下:第一章,我们给出本文的研究背景及一些预备知识.第二章,我们考虑由G-布朗运动驱动的反射倒向随机微分方程.我们采用不同于文献[48]的方法.具体来说,我们利用文献[76]推导出来的G-鞅表示定理,文献[16]得到的G-期望框架下的最优停止定理和文献[14]所介绍的方法.然而,我们也需要在G-期望框架下未被证明的G-上執表示定理,在本文中我们给出了证明.最终,我们证明了反射倒向随机微分方程的解的存在唯一性和一些估计.另外,我们得到了反射倒向随机微分方程的比较定理,这个比较定理是反射倒向方程理论中的一个有力的工具,在后面的章节中,也有应用.应该注意的是,在上面的方程中要求生成元f满足Lipschitz条件.在本章的第三部分,我们证明了当生成元f不满足Lipschitz条件时,方程至少存在一组解,推广了文献[48]的结论.第三章,我们考虑由G-布朗运动驱动的正倒向随机微分方程.在第一部分,通过迭代的方法,我们讨论由G-布朗运动驱动的完全耦合的正倒向随机微分方程,在参数满足单调性...
【文章页数】：126 页
【学位级别】：博士
【文章目录】：
摘要
Abstract
Chapter 1 Introduction
1.1 Backgrounds
1.2 Preliminaries
Chapter 2 Reflected backward stochastic differential equations driven byG-Brownian motion
2.1 Introduction and Preliminaries
2.2 Reflected backward stochastic differential equations driven by G-Brownianmotion
2.2.1 Some priori estimates and the uniqueness result
2.2.2 Existence of the solution
2.2.3 Comparison theorem
2.3 Reflected backward stochastic differential equations driven by G-Brownianmotion with continuous coefficients
Chapter 3 Forward-backward stochastic differential equations driven byG-Brownian motion
3.1 Fully coupled forward-backward stochastic differential equations drivenby G-Brownian motion
3.2 Reflected forward-backward stochastic differential equations driven byG-Brownian motion with continuous monotone coefficients
Chapter 4 Neutral stochastic partial functional integro-differential equa-tions driven by G-Brownian motion
4.1 Existence and uniqueness of the solution
4.2 Stability of the solution
4.3 An application
Chapter 5 Stochastic differential equations driven by G-Levy Process
5.1 Preliminaries
5.2 Stochastic differential equation driven by G-Levy Process
5.2.1 Exponential stability of the solution
5.2.2 An example
5.3 Existence of solution for stochastic differential equations driven by G-Levy process with discontinuous coefficients
Bibliography
Publications and Finished Papers
Acknowledgements
【参考文献】：
期刊论文
[1]p-th moment exponential stability of stochastic differential equations with impulse effect[J]. SHEN LiJuan 1,2 ＆ SUN JiTao 11 Department of Mathematics, Tongji University,Shanghai 200092,China;2 Department of Mathematics,Luoyang Normal University, Luoyang 471022,China. Science China（Information Sciences）. 2011(08)
[2]Some properties on G-evaluation and its applications to G-martingale decomposition[J]. SONG YongSheng Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Science China（Mathematics）. 2011(02)
[3]Survey on normal distributions,central limit theorem,Brownian motion and the related stochastic calculus under sublinear expectations[J]. PENG ShiGe Institute of Mathematics,Shandong University,Jinan 250100,China. Science in China（Series A:Mathematics）. 2009(07)
[4]非线性随机微分方程终值问题的适应解和连续依赖性[J]. 秦衍,夏宁茂,高焕超. 应用概率统计. 2007(03)
[5]NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS[J]. PENG Shige School of Mathematics and System Science, Shandong University, Jinan 250100, China.. Chinese Annals of Mathematics. 2005(02)
本文编号：3701373
【文章页数】：126 页
【学位级别】：博士
【文章目录】：
摘要
Abstract
Chapter 1 Introduction
1.1 Backgrounds
1.2 Preliminaries
Chapter 2 Reflected backward stochastic differential equations driven byG-Brownian motion
2.1 Introduction and Preliminaries
2.2 Reflected backward stochastic differential equations driven by G-Brownianmotion
2.2.1 Some priori estimates and the uniqueness result
2.2.2 Existence of the solution
2.2.3 Comparison theorem
2.3 Reflected backward stochastic differential equations driven by G-Brownianmotion with continuous coefficients
Chapter 3 Forward-backward stochastic differential equations driven byG-Brownian motion
3.1 Fully coupled forward-backward stochastic differential equations drivenby G-Brownian motion
3.2 Reflected forward-backward stochastic differential equations driven byG-Brownian motion with continuous monotone coefficients
Chapter 4 Neutral stochastic partial functional integro-differential equa-tions driven by G-Brownian motion
4.1 Existence and uniqueness of the solution
4.2 Stability of the solution
4.3 An application
Chapter 5 Stochastic differential equations driven by G-Levy Process
5.1 Preliminaries
5.2 Stochastic differential equation driven by G-Levy Process
5.2.1 Exponential stability of the solution
5.2.2 An example
5.3 Existence of solution for stochastic differential equations driven by G-Levy process with discontinuous coefficients
Bibliography
Publications and Finished Papers
Acknowledgements
【参考文献】：
期刊论文
[1]p-th moment exponential stability of stochastic differential equations with impulse effect[J]. SHEN LiJuan 1,2 ＆ SUN JiTao 11 Department of Mathematics, Tongji University,Shanghai 200092,China;2 Department of Mathematics,Luoyang Normal University, Luoyang 471022,China. Science China（Information Sciences）. 2011(08)
[2]Some properties on G-evaluation and its applications to G-martingale decomposition[J]. SONG YongSheng Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Science China（Mathematics）. 2011(02)
[3]Survey on normal distributions,central limit theorem,Brownian motion and the related stochastic calculus under sublinear expectations[J]. PENG ShiGe Institute of Mathematics,Shandong University,Jinan 250100,China. Science in China（Series A:Mathematics）. 2009(07)
[4]非线性随机微分方程终值问题的适应解和连续依赖性[J]. 秦衍,夏宁茂,高焕超. 应用概率统计. 2007(03)
[5]NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS[J]. PENG Shige School of Mathematics and System Science, Shandong University, Jinan 250100, China.. Chinese Annals of Mathematics. 2005(02)
本文编号：3701373
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