【作者】 任秀云；

【导师】 李娟；

【作者基本信息】 山东大学 ， 运筹学与控制论， 2013， 硕士

【摘要】 本文主要研究了两类平均场倒向随机微分方程的解的性质：带一致连续系数的平均场倒向随机微分方程解的性质以及带推广的一致连续系数的平均场倒向随机微分方程解的存在性。首先我们研究了一类带一致连续系数的平均场倒向随机微分方程。考虑形式如下的平均场倒向随机微分方程：我们对系数g作如下的假设：(B1)存在一个常数K>0,使得对于任意的t,y’,y,z,有(B2)对任意的y,z,g(t,y’,y,z)关于y’是非递减的；(B3)对所有的y’,y,z,(g(t,y’,y,z))I∈|0,T|∈H2(0,T;R);(B4)(一致连续条件)对于固定的t,g(t,.,·,.)是一致连续的,且关于t是一致的。也就是说,存在一个连续、次可加、非递减函数φ：R+→R+满足线性增长条件且φ(0)=0,且对所有的t∈[0,T],y1,y2∈R,z1,z2∈Rd,有：在上述假设条件下,我们通过构造一组满足Lipschitz条件的函数序列来逼近系数g,证明了当g不依赖y时,相应的平均场倒向随机微分方程的解是唯一的。而且还证明了使得带系数g+c的平均场倒向随机微分方程的解不唯一的实数c至多可数个。然后我们研究了一类带推广的一致连续系数的平均场倒向随机微分方程。令系数g满足下列假设条件：(H2) g(t,y’,y,z)关于y’非递减；(H3)(推广的一致连续条件)存在三个正的、确定的函数a(t),c(t)和d(t)满足:∫0τ[a(t)+c(t)+d2(t)]dt<∞,以及存在三个连续的、次可加、非递减函数φ1,φ2和ψ：R+→R+满足线性增长条件且φi(0)=0,i=1,2,ψ(0)=0,使得,对所有的t∈[0,T], y1,y2,y1’y2’∈R,z1,z2∈Rd,有：注意到,在这种情况下,系数g不一定满足线性增长条件。我们通过构造一组满足推广的Lipschitz条件的平均场倒向随机微分方程的Picard迭代序列来逼近原方程,证明了带推广的一致连续系数的平均场倒向随机微分方程解的存在性更多还原

【Abstract】 In this paper we study the properties of solutions of two types of mean-field backward stochastic differential equations (mean-field BSDEs):the prop-erties of solutions of the one-dimensional mean-field BSDEs with uniformly continuous coefficients and the existence of solutions of mean-field BSDEs with generalized uniformly continuous coefficients.Firstly we study mean-field BSDEs with uniformly continuous coefficients. We consider the mean-field BSDE as follows: We suppose that the coefficient g satisfies the following assumptions:(B1) there exists a constant K>0, such that, for any t, y’, y, z, we have(B2) for any y, z, g(t, y’,y, z) is nondecreasing in y’(B3) for any y’,y, z,(g(t, y’,y,z))t∈[0,T]∈H2(0, T; R);(B4)(Uniformly continuous condition) g(t,·,·,·) is uniformly continuous, uni-form with respect tot, that is, there exists a continuous, subadditive, nondecrea-sing function φ:R+→R+with linear growth and satisfying φ(0)=0such that, for any t∈[0,T], y1,y2∈R, z1,z2∈RdWe construct a sequence satisfying Lipschitz condition to approximate the coefficient g. We prove that, when g is independent of y, the associated mean-field BSDE has a unique solution. Moreover, the set of real numbers c where the mean-field BSDE with the coefficient g+c has a non-unique solution, is at most countable.Later we study mean-field BSDEs with generalized uniformly continuous coefficients. We suppose that the coefficient gsatisfies the following assumption-s:(H2) g(t, y’, y, z) is nondecreasing in y’(H3)(Generalized uniformly continuous condition) There exist three positive and deterministic functions a(t), c(t), d(t) satisfying∫02[a(t)+c(t)+d2(t)]dt<∞and three continuous, subadditive, nondecreasing functions φ1,φ2和ψ:R+→R+with linear growth and satisfying φ(0)=0,i=1,2, ψ(0)=0such that, for any t∈[0,T], y1, y2, y1’, y2’∈R, z1,z2∈Rd.Notice that with the conditions given above, g may not have a linear growth.We construct a Picard iterative sequences of mean-field BSDEs satisfying generalized Lipschitz condition to approximate the original equation. We prove the existence of solutions of mean-field BSDEs with generalized uniformly cont-inuous coefficients.更多还原