# Multi-dimensional Backward Stochastic Differential Equations of Diagonally Quadratic Generators with a Special Structure

## TL;DR

This paper establishes the well-posedness of multi-dimensional backward stochastic differential equations with diagonally quadratic generators, introducing new estimates and solvability conditions for small growth and triangular structures.

## Contribution

It provides new a priori estimates and proves existence and uniqueness of solutions for diagonally quadratic BSDEs with small off-diagonal growth and triangular structure.

## Key findings

- Unique solutions exist under small off-diagonal growth conditions.
- A new a priori estimate for diagonally quadratic BSDEs.
- Solvability results for triangular diagonally quadratic generators.

## Abstract

The present paper is devoted to the well-posedness of a type of multi-dimensional backward stochastic differential equations (BSDEs) with a diagonally quadratic generator. We give a new priori estimate, and prove that the BSDE admits a unique solution on a given interval when the generator has a sufficiently small growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a small growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$). Finally, we give a solvability result when the diagonally quadratic generator is triangular.