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

Abstract

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 $Z$.