超前倒向隨機微分方程解的存在唯一性及比較定理
發(fā)布時間：2018-12-19 20:18
【摘要】：本文主要研究多維超前倒向隨機微分方程(簡記為超前BSDE)平方可積解的存在唯一性定理和一維情形下解的比較定理,推廣了已有文獻中相關(guān)結(jié)果.第1章簡要介紹本文的研究背景、研究現(xiàn)狀、研究內(nèi)容及預備知識.第2章在生成元f關(guān)于Y及Y的超前項滿足一種特殊凹函數(shù)刻畫的非Lipschitz條件,關(guān)于Z及Z的超前項滿足對時間變量t不一致的Lipschitz條件下,通過皮卡迭代技術(shù),建立了該類多維超前BSDEs的平方可積解的存在唯一性定理(見定理2.1).隨后,我們通過介紹一個例子說明這一結(jié)論將Peng-Yang [2009],Yang-Robert [2013a],Wu-Wang [2012]中平方可積解的存在唯一性結(jié)果推廣到條件更一般情形.進一步,我們建立此條件下一維超前BSDEs解的比較定理(見定理2.2,定理2.3,定理2.4,定理2.5),推廣Peng-Yang [2009],Xu [2011], Wu-Wang [2012]和Zhang [2014]中相關(guān)的比較定理結(jié)果.第3章在生成元f關(guān)于Y及Y的超前項滿足Osgood條件,關(guān)于Z及Z的超前項滿足一致Lipschitz條件下,通過構(gòu)造多個一致連續(xù)函數(shù)的一致逼近的Lipschitz函數(shù)序列,建立了該類多維超前BSDEs的平方可積解的存在唯一性定理(見定理3.1),進一步豐富了超前BSDEs解的存在唯一性研究成果.第4章,我們對本文進行了簡單總結(jié)及對接下來研究工作的展望.
[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.
【學位授予單位】：中國礦業(yè)大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O211.63
，
本文編號：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.
【學位授予單位】：中國礦業(yè)大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O211.63
，
本文編號：2387394
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