拟单调增长连续的多维反射正倒向随机微分方程

发布时间：2018-02-27 14:09

  本文关键词： 反射正倒向随机微分方程 多维 拟单调增长 比较定理 逼近　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文研究的是多维反射正倒向随机微分方程(简记为反射FBSDEs).运用多维反射倒向随机微分方程(简记为反射BSDEs)解的存在唯一性、比较定理和"四步法",在系数满足拟单调增长连续的条件下证明了多维反射FBSDEs解的存在性。1990年,Pardoux和Peng首先在[2]中给出了如下的非线性BSDE和解的存在唯一性定理:近20年来,BSDE作为研究工程控制、系统科学、随机控制和金融数学等方面的理论工具,被越来越多的人所熟知。1993年,Antonelli[21]在研究控制学理论时首先提出了正倒向随机微分方程(简记为FBSDE),他给出了 FBSDE在系数满足Lipschitz条件下,解的存在唯一性定理。1994年,Ma,Protter和Yong[22]利用研究偏微分方程(简记为PDE)系统的方法,给出了求解FBSDE的"四步法",该方法使得随机控制理论和PDE理论完美结合,为解决数理金融等方面的问题提供了方法。1997年,El-Karouietal.[13]首次提出了一维反射 BSDE,并给出了 Lipschitz条件下解的存在唯一性定理和比较定理。2010年,Huang,Lepeltier和Wu[27]在Antonelli和Hamadene[25]给出的一类完全耦合的FBSDE的研究基础上,做出了延伸,加入了障碍过程进行约束,从而得到了一维反射FBSDE,并给出了生成元满足单调连续条件时解的存在性。2010年,Wu和Xiao[26]给出了多维反射BSDEs解的存在唯一性定理和比较定理。2012年,El.Asri[28]研究了一类多维反射FBSDEs,并给出了在最优停时问题上的应用。2013年,Aazizi和Fakhouri[29]研究了斜反射和无界停时的多维FBSDEs.在Huang,Lepeltier和Wu[27]给出的一维反射FBSDE的基础上,我们可以很自然的提出几个疑问,如何构造多维反射FBSDEs的理论框架?如何证明多维反射FBSDEs在系数满足拟单调增长连续的条件下解的存在性?本文共分为四个章节。第一章:引言,介绍前人在SDE、BSDE、反射BSDE、FBSDE、反射FBSDE等方面所做的研究,提出我们所研究的课题,叙述本文的结构框架。第二章:受Huang,Lepeltier和Wu[27]中一维反射FBSDE的指点,我们建立了多维反射FBSDEs在拟单调增长连续条件下的理论模型,并为证明做出相应的前期准备。首先给出如下多维反射FBSDEs模型:在前期准备方面我们给出了多维SDEs,多维BSDEs和多维反射BSDEs的比较定理及函数逼近的相关知识。第三章:给出反射正倒向随机微分方程在拟单调增长连续条件下解的存在性定理。我们假设(2.1)中的系数和参数满足如下假设(ⅰ)6是关于y单调增长,关于x拟单调增长的函数;(ⅱ)f是关于x单调增长,关于y拟单调增长的函数;f的第j行分量f_j只含有z的第j行元素z_j,f_j和每一个z_l,l≠j是相互独立的;(ⅲ)存在一个常数C≥0使得为了得到我们的证明,我们先通过方程(2.1)构造迭代数列参照Ma,Prottcr和Yong[22],我们通过"四步法",应用迭代算法和逼近技术证明了解的存在性。第四章:对我们的研究成果进行了总结,并对进一步的研究做出了展望。
[Abstract]:In this paper, we discuss the existence and uniqueness of the solution of the multidimensional reflection forward backward stochastic differential equation (abbreviated as the reflected FBSD eschus) by using the multidimensional reflection backward stochastic differential equation (abbreviated as the reflection BSD Ess). The comparison theorem and "four-step method" prove the existence of multi-dimensional reflection FBSDEs solution under the condition that the coefficients satisfy the condition of quasi-monotone growth continuity. In 1990, we first gave the existence and uniqueness theorem of nonlinear BSDE solution in [2]. In the past 20 years, BSDE has been used as research engineering control. Theoretical tools in systems science, stochastic control and financial mathematics, In 1993, when studying the theory of control, Antonelli put forward the forward backward stochastic differential equation (FBSDE). He gave FBSDE under the condition that the coefficient satisfies the Lipschitz condition. In 1994, by using the method of studying partial differential equations (abbreviated as PDE), a four-step method for solving FBSDE is given. This method combines stochastic control theory with PDE theory perfectly. In 1997, El-Karouietal.in 1997, El-Karouietal. [13] proposed the one-dimensional reflection BSDEfor the first time, and gave the existence and uniqueness theorems and comparison theorems of solutions under Lipschitz condition. In 2010, Huang Lepeltier and Wu [27] gave a class of endings in Antonelli and Hamadene [25]. Based on the research of fully coupled FBSDE, An extension is made, a constraint is added to the barrier process, In 2010, Wu and Xiao [26] gave the existence and uniqueness theorem of multidimensional reflection BSDEs solution. In 2012, El.Asri [28] studied a kind of multidimensional reflection BSDEs solution. In 2013, Fakhouri and Aazizi studied the multi-dimensional FBSDE of oblique reflection and unbounded stopping time. On the basis of the one-dimensional reflection FBSDE given by Huang Li Lepeltier and Wu [27], Naturally, we can ask a few questions, how to construct the theoretical framework of multidimensional reflection FBSDEs? How to prove the existence of solutions of multi-dimensional reflection FBSDEs under the condition that the coefficients satisfy the condition of quasi-monotone growth continuity? This paper is divided into four chapters. Chapter 1: introduction, introduce the previous research on SDE / BSDE, reflective BSDE / FBSDE, reflect FBSDE and so on, put forward our research topic, and describe the structure of this paper. Chapter 2: the reference of one-dimensional reflection FBSDE by Huang Li Lepeltier and Wu [27]. In this paper, we establish a theoretical model of multi-dimensional reflection FBSDEs under the condition of quasi-monotone growth. In order to prove the corresponding preliminary preparation, the following multi-dimensional reflective FBSDEs model is given: in the aspect of prepreparation, we give the comparison theorem of multidimensional SDES, multidimensional BSDEs and multidimensional reflection BSDEs and the related knowledge of function approximation. Chapter: we give the existence theorem of the solution of the reflected forward backward stochastic differential equation under the condition of quasi-monotone growth. We assume that the coefficients and parameters in the reflection forward backward stochastic differential equation satisfy the following assumptions (I ~ (6) is about y monotone growth, On the function of x quasi monotone growth (鈪，

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