Mean-field Backward Stochastic Differential Equations with Quadratic Growth

Abstract

In this paper we initially investigate one dimensional mean-field backward stochastic differential equations (MFBS-DEs) with coefficients ${f}$ quadratic in ${z}$ under different conditions. We first prove the existence of solutions of this MFBSDEs with bounded terminal value under two cases: The first case is that ${f}$ is continuous, non-decreasing in $y^{\prime}$, and is of super-linear growth in ${y}$ and quadratic growth in ${z}$; the second one is that ${f}$ is monotonic in ${y}$, non-decreasing in $y^{\prime}$ and quadratic in ${z}$. And then, we extend the above two cases to the general situation, i.e., $f$ is continuous, non-decreasing in $y^{\prime}$ and of linear growth in $(y^{\prime},\,y)$, as well as quadratic in $z$.