Multi-dimensional Mean-field Type Backward Stochastic Differential Equations with Diagonally Quadratic Generators

TL;DR

This paper investigates multi-dimensional mean-field backward stochastic differential equations with diagonally quadratic generators, establishing local and global existence and uniqueness results under specific growth conditions.

Contribution

It introduces new methods for proving existence and uniqueness of solutions for complex BSDEs with mean-field interactions and diagonally quadratic growth.

Findings

Proved local existence and uniqueness for diagonally quadratic generators.

Established global solutions under logarithmic growth conditions.

Developed new a priori estimates for these equations.

Abstract

In this paper, we study the multi-dimensional backward stochastic differential equations (BSDEs) whose generator depends also on the mean of both variables. When the generator is diagonally quadratic, we prove that the BSDE admits a unique local solution with a fixed point argument. When the generator has a logarithmic growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a logarithmic growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$), we give a new apriori estimate and obtain the existence and uniqueness of the global solution.