Mean-field Backward Stochastic Differential Equations With Uniformly Continuous Coefficients And Its Applications

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+â†'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,φ2å'ŒÏˆ:R+â†'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.