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 ? satisfies two stochastic assumptions,respectively:Where ? is FT-rmeasurable;Q is an IF-ada,pted increa,sing continuous process:?�[0,T]?R.Qo = 0.Assumption 1:? satisfies the following stochastic Lipschitz conditions(H1)F-progressively measurable stochastic process ?(�,0,,0,0,0)is bounded.(H2)There exist three IF-progressively measurable stochastic processes L,l and ?:? �[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.:|?(t,y1,z1,y'1,z'1)-?(t,y2,z2,y'2,z'2)|?Lt(|y1-y2|+ |y'1-y'2)+?tlt(|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:? satisfies above(H1)and other stochastic monotonic conditions:(H4)There exists a constant M and F-progressively measurable stochastic processes:?:? �[0,T]? R,l ?:�[0,T]? R+,such that?tdQt = dt,(?)(H5)For all y1,y'1,y2,y'2?Rm,z1,z'1,z2,z'2?Rm�k,dP(?)dQ-a.e.? is continuous in(y,y'),monotonic inn y,and Lipschitz in y':<y1-y2?(t,y1,z1,y'1,z'1)-?(t,y2,z1,y'1,z'1)>??t|y1-y2|2,|?(t,y1,z1,y'1,z'1)-?(t,y2,z1,y'1,z'1)|??t|y'1-y'2|;? is Lipschitz continuous in(z,z'):|?(t,y1,z1,y'1,z'1)-?(t,y1,z2,y'1,z'2)? ?tlt(|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.