General Mean-field Backward Stochastic Differential Equations

This paper studies mainly the properties of solution for mean-field backward s-tochastic differential equations under Lipschitz condition and continuous condition;the existence of Lp(1<P≤2)solutions under continuous condition and uniqueness of Lp(1<P≤2)solution under uniformly continuous condition.And we first prove the comparison theorem under Lipschitz condition.Considering the following mean-field backward stochastic differential equations where ξ∈L2(FT;R).(H3.1)(f(s,δ0,0,0))s∈[O,T]∈HF2(O,T;R).(H3.2)f is Lipschitz,that is,there exists a constant C∈R+,such that for allμ,μ’∈P2(RxRd),y,y’∈R z,z’ ∈Rd,Under(H3.1)and(H3.2),we prove equation(1)has a unique solution by using the iteration.There exists counterexamples to show that if the driver f depends on the law of Z or f is non-increasing with respect to μ we can’t get the comparison theorem.Therefore,we give the comparison theorem when the driver f doesn’t depend on the law of Z:The comparison theorem plays an important role in the following proof.(H4.1)Linear growth:there exists a constant K≥0,such that(H4.3)f(s,ω,μ,y,z)is continuous in(y,z)and there is a continuous and increasing Here p(0+)= 0.Under(H4.1)-(H4.3),we prove equation(2)exists solutions with the help of priori estimate and Lipschitz function approximating continuous functions.(H5.2.1)For all(s,ω,μ,y,z),｜f(s,ω,μ,y,z)｜≤ K(1+Wp+(μ,δ0)+ ｜y｜+ ｜z｜).(H5.2.2)Terminal condition ξ∈ Lp.(H5.2.3)f(s,ω,μ,y,z)is continuous in(y,z)and there is a continuous and increas-Here ρ(0+)=0.Under(H4.2),(H5.2.1)-(H5.2.3),we prove the existence of LP solutions for equation(2)by using generalized Ito formula and Lipschitz function approximates continuous functions.(H5.3.1)f is uniformly continuous in y uniformly with(s,ω,μ,Z).(H5.3.2)f is uniformly continuous in z uniformly with(s,ω,μ,y).(H5.3.3)f is uniformly continuous in μ uniformly with(s,ω,μ,z).Under(H4.2),(H5.2.1),(H5.2.2)and(H5.3.1)-(H5.3.3),we prove the existence and uniqueness of LP solution for equation(2)by using recursion equation.