超前倒向随机微分方程解的存在唯一性及比较定理
发布时间：2018-12-19 20:18
【摘要】：本文主要研究多维超前倒向随机微分方程(简记为超前BSDE)平方可积解的存在唯一性定理和一维情形下解的比较定理,推广了已有文献中相关结果.第1章简要介绍本文的研究背景、研究现状、研究内容及预备知识.第2章在生成元f关于Y及Y的超前项满足一种特殊凹函数刻画的非Lipschitz条件,关于Z及Z的超前项满足对时间变量t不一致的Lipschitz条件下,通过皮卡迭代技术,建立了该类多维超前BSDEs的平方可积解的存在唯一性定理(见定理2.1).随后,我们通过介绍一个例子说明这一结论将Peng-Yang [2009],Yang-Robert [2013a],Wu-Wang [2012]中平方可积解的存在唯一性结果推广到条件更一般情形.进一步,我们建立此条件下一维超前BSDEs解的比较定理(见定理2.2,定理2.3,定理2.4,定理2.5),推广Peng-Yang [2009],Xu [2011], Wu-Wang [2012]和Zhang [2014]中相关的比较定理结果.第3章在生成元f关于Y及Y的超前项满足Osgood条件,关于Z及Z的超前项满足一致Lipschitz条件下,通过构造多个一致连续函数的一致逼近的Lipschitz函数序列,建立了该类多维超前BSDEs的平方可积解的存在唯一性定理(见定理3.1),进一步丰富了超前BSDEs解的存在唯一性研究成果.第4章,我们对本文进行了简单总结及对接下来研究工作的展望.
[Abstract]:In this paper, we mainly study the existence and uniqueness theorem of square integrable solutions of multidimensional backward stochastic differential equations (abbreviated as advanced BSDE) and the comparison theorem of solutions in one-dimensional cases, which generalize the relevant results in the literature. Chapter 1 briefly introduces the research background, research status, research content and preparatory knowledge. In chapter 2, under the condition that the leading term of generator f satisfies a special concave function characterizing non-Lipschitz condition, and the leading term of Z and Z satisfies the Lipschitz condition which is inconsistent with the time variable t, the Pika iteration technique is adopted. The existence and uniqueness theorem of square integrable solution for this kind of multidimensional advanced BSDEs is established (see Theorem 2.1). Then, we introduce an example to show that the existence and uniqueness results of square integrable solutions in Peng-Yang [2009], Yang-Robert [2013a], Wu-Wang [2012] are generalized to more general conditions. Furthermore, we establish a comparison theorem for one-dimensional advanced BSDEs solutions under this condition (see Theorem 2.2, Theorem 2.3, Theorem 2.4, Theorem 2.5), which generalizes Peng-Yang [2009], Xu [2011], The results of comparison theorems in Wu-Wang [2012] and Zhang [2014]. In chapter 3, under the condition that the leading term of generator f satisfies the Osgood condition for Y and Y, and the leading term for Z and Z satisfies the uniform Lipschitz condition, a sequence of uniformly approximated Lipschitz functions of several uniformly continuous functions is constructed. The existence and uniqueness theorem of the square integrable solution of this kind of multidimensional advanced BSDEs is established (see Theorem 3.1), which further enriches the research results of the existence and uniqueness of the leading BSDEs solution. In chapter 4, we make a brief summary of this paper and look forward to the future research work.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O211.63
，
本文编号：2387394
[Abstract]:In this paper, we mainly study the existence and uniqueness theorem of square integrable solutions of multidimensional backward stochastic differential equations (abbreviated as advanced BSDE) and the comparison theorem of solutions in one-dimensional cases, which generalize the relevant results in the literature. Chapter 1 briefly introduces the research background, research status, research content and preparatory knowledge. In chapter 2, under the condition that the leading term of generator f satisfies a special concave function characterizing non-Lipschitz condition, and the leading term of Z and Z satisfies the Lipschitz condition which is inconsistent with the time variable t, the Pika iteration technique is adopted. The existence and uniqueness theorem of square integrable solution for this kind of multidimensional advanced BSDEs is established (see Theorem 2.1). Then, we introduce an example to show that the existence and uniqueness results of square integrable solutions in Peng-Yang [2009], Yang-Robert [2013a], Wu-Wang [2012] are generalized to more general conditions. Furthermore, we establish a comparison theorem for one-dimensional advanced BSDEs solutions under this condition (see Theorem 2.2, Theorem 2.3, Theorem 2.4, Theorem 2.5), which generalizes Peng-Yang [2009], Xu [2011], The results of comparison theorems in Wu-Wang [2012] and Zhang [2014]. In chapter 3, under the condition that the leading term of generator f satisfies the Osgood condition for Y and Y, and the leading term for Z and Z satisfies the uniform Lipschitz condition, a sequence of uniformly approximated Lipschitz functions of several uniformly continuous functions is constructed. The existence and uniqueness theorem of the square integrable solution of this kind of multidimensional advanced BSDEs is established (see Theorem 3.1), which further enriches the research results of the existence and uniqueness of the leading BSDEs solution. In chapter 4, we make a brief summary of this paper and look forward to the future research work.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O211.63
，
本文编号：2387394
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