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

TL;DR

This paper establishes comparison theorems for one-dimensional mean-field BSDEs with coefficients depending on the joint law of the solution, including the $Z$ component, using Malliavin calculus and BMO martingale techniques.

Contribution

It introduces new comparison theorems for mean-field BSDEs where coefficients depend on the joint law of $(Y,Z)$, including the $Z$ component, which was previously not well-understood.

Findings

Established comparison theorems for mean-field BSDEs with joint law dependence.

Compared both $Y$ and $Z$ components of solutions.

Provided strong comparison results under new conditions.

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 $Z$.