Mean-Field Backward Stochastic Differential Equations With Stochastic Coefficients

We mainly study the existence and the uniqueness of the solution and related prop-erties of mean-field backward stochastic differential equations(mean-field BSDEs)whose generator Φ satisfies two stochastic assumptions,respectively:Where η is FT-rmeasurable;Q is an IF-ada,pted increa,sing continuous process:Ω×[0,T]→R.Qo = 0.Assumption 1:Φ satisfies the following stochastic Lipschitz conditions(H1)F-progressively measurable stochastic process Φ(·,0,,0,0,0)is bounded.(H2)There exist three IF-progressively measurable stochastic processes L,l and α:Ω ×[0,T]→ R+;such that(H3)For all t∈[0,T]y1,y’1,y2,y’2∈Rm,z1,z’1,z2,z’2∈m×k,P-a.s.:|Φ(t,y1,z1,y’1,z’1)-Φ(t,y2,z2,y’2,z’2)|≤Lt(|y1-y2|+ |y’1-y’2)+αtlt(|z1-z2| + |z’1-z’2|).We prove the existence and the uniqueness of the solution for mean-field BSDE(0.0.1)under Assumption 1 by using a priori estimates,approximation,convergence and the backward Stielt.jes-Gronwall inequality.Later we prove mean-field BSDE(0.0.1)sat-isfies the multidimensional comparison theorem.Assumption 2:Φ satisfies above(H1)and other stochastic monotonic conditions:(H4)There exists a constant M and F-progressively measurable stochastic processes:μ:Ω ×[0,T]→ R,l α:×[0,T]→ R+,such thatαtdQt = dt,(?)(H5)For all y1,y’1,y2,y’2∈Rm,z1,z’1,z2,z’2∈Rm×k,dP(?)dQ-a.e.Φ is continuous in(y,y’),monotonic inn y,and Lipschitz in y’:<y1-y2Φ(t,y1,z1,y’1,z’1)-Φ(t,y2,z1,y’1,z’1)>≤μt|y1-y2|2,|Φ(t,y1,z1,y’1,z’1)-Φ(t,y2,z1,y’1,z’1)|≤μt|y’1-y’2|;Φ is Lipschitz continuous in(z,z’):|Φ(t,y1,z1,y’1,z’1)-Φ(t,y1,z2,y’1,z’2)≤ αtlt(|z1-z2|+|z’1-z’2|).We prove the existence and the uniqueness of the solution of mean-field BSDE(0.0.1)under Assumption 2.mean-field BSDE(0.0.1)also have the multidimensional comparison theorem.