【作者】 肖华；

【导师】 吴臻；

【作者基本信息】 山东大学 ， 概率论与数理统计， 2005， 硕士

【摘要】 本文研究的是多维反射倒向随机微分方程（简记为BSDE）解的存在唯一性,比较定理及其应用。 众所周知,BSDE是一个新兴的研究方向,它的出现为研究金融数学,随机最优控制及偏微分方程等问题提供了有利的工具。如下的非线性BSDE-dY（t）=f（t,Y（t）,Z（t））dt-Z（t）dB_t,Y_T=ξ是由Pardoux和Peng于1990年在[1]中首先介绍的,后来Peng于1992年在[2]中证明了一维BSDE的比较定理,周海滨于1999年在[3]中证明了一类多维BSDE的比较定理,证明方法是构造了一个特殊的函数,这个函数是Buckdahn和Peng于1999年在[4]中首次介绍的。周海滨还将多维BSDE比较定理应用于证明多维拟单调连续系数BSDE解的存在性。El. Karoui et al在[5]中研究了带一个障碍的一维反射BSDE,给出了一维情况下解的存在唯一性定理和比较定理,同时还在Markov框架下研究了一维反射BSDE与非线性抛物型偏微分方程的联系。 我们现在很自然的提出,如何建立多维反射BSDE的框架,建立之后,多维反射BSDE的解是否存在唯一,解的比较定理是否成立,是否也可以将多维反射BSDE的比较定理应用于证明多维拟单调连续系数反射BSDE解的存在性? 本文共分四章。 第一章:引言,叙述前人所作的工作以及问题的由来。 第二章:受El. Karoui et al[5]中一维反射BSDE模型的启发,我们提出了如下的多维反射BSDE模型: 首先假定 （H2.1）ξ∈L_n²。 （H2.2）f:Ω×[0,T]×Rⁿ×R^n×d→Rⁿ,（y,z）∈Rⁿ×R^n×d,f（·,y,z）∈H_n²。 （H2.3）|f（t,y,z）-f（t,y′,z′）|≤C（|y-y′|+|z-z′|）,C>0,y,y′∈Rⁿ,z,z′∈R^n×d。 给出一个n维的障碍{S（t）,0≤t≤T}∈Rⁿ满足 （H2.4）{S（t）,0≤t≤T}∈Rⁿ是一个连续的循序可测的Rⁿ中的过程,并且满足E（（?）|S+（t）|²）<+∞,S（T）≤ξ。更多还原

【Abstract】 In this paper, we study the multi-dimensional reflected backward stochastic differential equations （BSDE in short）. The existence and uniqueness result of the solution for this kind of equation was proved, and we also give one kind of multi-dimensional comparison theorem for the reflected BSDE and then use the comparison theorem as the tool to prove one existence result for multi-dimensional reflected BSDE where the coefficient is continuous and has the linear growth.It is well known that BSDE has become a field of increasing activity. It is becoming an important tool in study of fanancial mathematics, stochastic optimal control problems and partial defferential equation. The following nonlinear BSDEwas first introduced by Pardoux and Peng in 1990 （see [1]）. After that, Peng proved the comparison theorem for one-dimensional BSDE in 1992 （see [2]）. Zhou proved one kind of comparison theorem for mul-dimensional BSDE in 1999 （see [3]）, and then use the comparison theorem as the tool to prove one existence result for multi-dimensional BSDE where the coefficient is continuous and has the linear growth. El.Karoui et al studied one-dimensional reflected BSDE with one barrier in 1997 （see [5]）, and then proved the existence and uniqueness result and comparison theorem of the solution for this kind of equation.We divide this paper into four chapters. The first chapter is an introduction . In chapter 2, we give the following model for multi-dimensional reflected BSDE. We first give the comparison definition for two vectors in Rⁿ:and then assume:and we give one n-dimensional obstacle {S（t）,0 ≤ t ≤ T} ∈ Rⁿ satisfying （H2.4） {S{t),0 ≤ t ≤ T} ∈ Rⁿ is a continuous progressively measurable Rⁿ-valued process satisfying E（ sup |S+（t）|²） < +∞,S{T) ≤ ξ.Here S+（t） is a Rⁿ vector , the jth element is （S_j⁺（t））.（f, ξ, S） is called one group of standard parameter for n-dimensional reflected BSDE, if it satisfies （H2.1） - （H2.4） .And then we call {（Y（t）, Z（t）, K（t））,0 ≤ t ≤ T} to be the solution for n-dimensional reflected BSDE if it satisfiesincrease process satisfying K_j（0） = 0 and ∫₀^T （Y_j（t） — S_j（t）） dK_j（t） = 0,Theorem 2.1. We assume （f, ξ,S） satisfies （H2.1） — （H2.4）, then there exists a group of solution （Y,Z,K） for n-dimensional reflected BSDE satisfying （H2.5） - （H2.8）.Theorem 2.2. We assume （f, ξ,S） satisfies （H2.1） - （H2.4）, then there exists at most a group of solution （Y, Z, K） for n-dimensional reflected BSDE satisfying （H2.5） —（H2.8）.In chapter 3, we getTheorem 3.1. Let （f¹, ξ¹,S¹） and （f²,ξ2, S²） be two standard parameters of the n-dimensional reflected BSDE satisfying （H2.1） - （H2.4）, and suppose in addition the followingLet （Y¹,Z¹,K¹） and （Y²,Z²,K²） be the solution respective to （f¹,ξ¹,S¹） and （f²,ξ²,S²）, then Y¹（t） ≤ Y²{t).We notice that the condition （ii） of Theorem 3.1 , i.e. the quasi-monotonously in-creasing assumption , is different to the common monotonous assumption for the generator of the reflected BSDE . Our question is whether it is possible to change to the weaker assumption :（n’）fj（t,y,zl）<ff（t,y,z2）, z) = zjj = 1,2, ? ? ? ,n.From the counterexample 3.1, we know that the comparison theorem does not hold under the assumption （ii’）.In chapter 4, we extend the existence result of the solution for the multi-dimensional reflected BSDE in chapter 2. Using the comparison Theorem 3.1 as the main tool and the suitable approximation of the generator coefficient, we prove the existence for n-dimensional reflected BSDE where the coefficient is continuous and has linear growth.Now we assume f（t, u, y, z） :[0,T]xOxRnx Rnxd —> Rn be a measurable mapping and satisfy the following（H4.1） the j-th line fj of / only contains the j-th element of z,i.e.3g{t,ui,y,j) : [0,T] x ft x Rn x Rd —> Rn,s.t.fj（t,u,y,z） = gj（t,u,y,Zj）,Vt€ [0,T],u; € U,y € Rn,z e Rnxd; （H4.2） linear growth : 3 Co > 0, si.\f{t,u,y,z)\ < C0（l + \y\ + \z\）,V t € [0,T),u 6 Q,y?nxd（H4.3） for fixed t,oj,f（t,u,-,-） is continuous .（H4.4） for fixed t,u,fj（t,u,-,z） is quasi-monotonously increasing, that is for j = l,2,...,n,fj{t,u,y\z) < fj{t,u,y2,z),Vy1,y2 G Rn, y) = y],y} < y2,1 ± j-And then we haveTheorem 4.1. Assume / satisfies （H4.1）-（H4.4） and ￡ € L￡, S{t) 6 S’, then the n-dimensional reflected BSDE exists a triple of solution （Y, Z, K） satisfying （H2.5） - （H2.8）.更多还原