在本文中,我們討論G－期望框架下由G-布朗運動和G-Levy過程驅(qū)動的幾類隨機微分方程.論文由五個部分組成,結構如下:第一章,我們給出本文的研究背景及一些預備知識.第二章,我們考慮由G-布朗運動驅(qū)動的反射倒向隨機微分方程.我們采用不同于文獻[48]的方法.具體來說,我們利用文獻[76]推導出來的G-鞅表示定理,文獻[16]得到的G-期望框架下的最優(yōu)停止定理和文獻[14]所介紹的方法.然而,我們也需要在G-期望框架下未被證明的G-上執(zhí)表示定理,在本文中我們給出了證明.最終,我們證明了反射倒向隨機微分方程的解的存在唯一性和一些估計.另外,我們得到了反射倒向隨機微分方程的比較定理,這個比較定理是反射倒向方程理論中的一個有力的工具,在后面的章節(jié)中,也有應用.應該注意的是,在上面的方程中要求生成元f滿足Lipschitz條件.在本章的第三部分,我們證明了當生成元f不滿足Lipschitz條件時,方程至少存在一組解,推廣了文獻[48]的結論.第三章,我們考慮由G-布朗運動驅(qū)動的正倒向隨機微分方程.在第一部分,通過迭代的方法,我們討論由G-布朗運動驅(qū)動的完全耦合的正倒向隨機微分方程,在參數(shù)滿足單調(diào)性... 

【文章頁數(shù)】：126 頁

【學位級別】：博士

【文章目錄】：
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


【參考文獻】：
期刊論文
[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]非線性隨機微分方程終值問題的適應解和連續(xù)依賴性[J]. 秦衍,夏寧茂,高煥超.  應用概率統(tǒng)計. 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)



本文編號：3701373