本文關(guān)鍵詞： 反射正倒向隨機(jī)微分方程 多維 擬單調(diào)增長 比較定理 逼近　出處：《山東大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：本文研究的是多維反射正倒向隨機(jī)微分方程(簡記為反射FBSDEs).運(yùn)用多維反射倒向隨機(jī)微分方程(簡記為反射BSDEs)解的存在唯一性、比較定理和"四步法",在系數(shù)滿足擬單調(diào)增長連續(xù)的條件下證明了多維反射FBSDEs解的存在性。1990年,Pardoux和Peng首先在[2]中給出了如下的非線性BSDE和解的存在唯一性定理:近20年來,BSDE作為研究工程控制、系統(tǒng)科學(xué)、隨機(jī)控制和金融數(shù)學(xué)等方面的理論工具,被越來越多的人所熟知。1993年,Antonelli[21]在研究控制學(xué)理論時(shí)首先提出了正倒向隨機(jī)微分方程(簡記為FBSDE),他給出了 FBSDE在系數(shù)滿足Lipschitz條件下,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(簡記為PDE)系統(tǒng)的方法,給出了求解FBSDE的"四步法",該方法使得隨機(jī)控制理論和PDE理論完美結(jié)合,為解決數(shù)理金融等方面的問題提供了方法。1997年,El-Karouietal.[13]首次提出了一維反射 BSDE,并給出了 Lipschitz條件下解的存在唯一性定理和比較定理。2010年,Huang,Lepeltier和Wu[27]在Antonelli和Hamadene[25]給出的一類完全耦合的FBSDE的研究基礎(chǔ)上,做出了延伸,加入了障礙過程進(jìn)行約束,從而得到了一維反射FBSDE,并給出了生成元滿足單調(diào)連續(xù)條件時(shí)解的存在性。2010年,Wu和Xiao[26]給出了多維反射BSDEs解的存在唯一性定理和比較定理。2012年,El.Asri[28]研究了一類多維反射FBSDEs,并給出了在最優(yōu)停時(shí)問題上的應(yīng)用。2013年,Aazizi和Fakhouri[29]研究了斜反射和無界停時(shí)的多維FBSDEs.在Huang,Lepeltier和Wu[27]給出的一維反射FBSDE的基礎(chǔ)上,我們可以很自然的提出幾個(gè)疑問,如何構(gòu)造多維反射FBSDEs的理論框架?如何證明多維反射FBSDEs在系數(shù)滿足擬單調(diào)增長連續(xù)的條件下解的存在性?本文共分為四個(gè)章節(jié)。第一章:引言,介紹前人在SDE、BSDE、反射BSDE、FBSDE、反射FBSDE等方面所做的研究,提出我們所研究的課題,敘述本文的結(jié)構(gòu)框架。第二章:受Huang,Lepeltier和Wu[27]中一維反射FBSDE的指點(diǎn),我們建立了多維反射FBSDEs在擬單調(diào)增長連續(xù)條件下的理論模型,并為證明做出相應(yīng)的前期準(zhǔn)備。首先給出如下多維反射FBSDEs模型:在前期準(zhǔn)備方面我們給出了多維SDEs,多維BSDEs和多維反射BSDEs的比較定理及函數(shù)逼近的相關(guān)知識(shí)。第三章:給出反射正倒向隨機(jī)微分方程在擬單調(diào)增長連續(xù)條件下解的存在性定理。我們假設(shè)(2.1)中的系數(shù)和參數(shù)滿足如下假設(shè)(ⅰ)6是關(guān)于y單調(diào)增長,關(guān)于x擬單調(diào)增長的函數(shù);(ⅱ)f是關(guān)于x單調(diào)增長,關(guān)于y擬單調(diào)增長的函數(shù);f的第j行分量f_j只含有z的第j行元素z_j,f_j和每一個(gè)z_l,l≠j是相互獨(dú)立的;(ⅲ)存在一個(gè)常數(shù)C≥0使得為了得到我們的證明,我們先通過方程(2.1)構(gòu)造迭代數(shù)列參照Ma,Prottcr和Yong[22],我們通過"四步法",應(yīng)用迭代算法和逼近技術(shù)證明了解的存在性。第四章:對我們的研究成果進(jìn)行了總結(jié),并對進(jìn)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，

本文編號(hào)：1542989