# Quadratic BSDEs with mean reflection

## TL;DR

This paper establishes the existence and uniqueness of solutions for quadratic backward stochastic differential equations with mean reflection, extending previous work to handle quadratic growth in the generator.

## Contribution

It introduces a method to prove well-posedness of quadratic BSDEs with mean reflection, including local and global solutions under bounded terminal conditions.

## Key findings

- Unique local solutions for quadratic BSDEs with mean reflection.
- Global solutions constructed by stitching local solutions.
- Applicability to super-hedging problems under risk constraints.

## Abstract

The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the $z$ argument. This work is the sequel of Briand et al. [BSDEs with mean reflection, arXiv:1605.06301] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the $y$ argument.