Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution (Y,Z)

For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of Z-component of the solution process (Y,Z). A natural question is whether general mean-field BSDEs whose coefficients depend on the law of Z have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process (Y,Z). With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component Y of the solution (Y,Z) for such mean-field BSDEs, but also the second component Z. After a discussion of mean-field BSDEs whose terminal condition and the driving coefficient are Malliavin differentiable, the results are extended in a second phase to the case without assumption of Malliavin differentiability.