Multi-dimensional non-Markovian backward stochastic differential equations of interactively quadratic generators

TL;DR

This paper establishes existence and uniqueness results for multi-dimensional non-Markovian BSDEs with interactively quadratic generators, advancing the understanding of their solvability under various growth conditions.

Contribution

It introduces new conditions for local and global solutions of non-Markovian BSDEs with complex quadratic growth, improving upon prior research and addressing open problems.

Findings

Proved local existence and uniqueness of bounded solutions.

Established global solutions under broader growth conditions.

Provided examples including partial answers to open problems.

Abstract

This paper is devoted to a general solvability of multi-dimensional non-Markovian backward stochastic differential equations (BSDEs) with interactively quadratic generators. Some general structures of the generator $g$ are posed for both local and global existence and uniqueness results on BSDEs, which admit a general growth of the generator $g$ in the state variable $y$, and a quadratic growth of the $i$th component $g^i$ both in the $j$th row $z^j$ of the state variable $z$ for $j\neq i$ (by which we mean the ``{\it interactively quadratic}" growth) and in the $i$th row $z^i$ of $z$. We first establish an existence and uniqueness result on local bounded solutions and then several existence and uniqueness results on global bounded and unbounded solutions. They improve several existing works in the non-Markovian setting, and also incorporate some interesting examples, one of which is a partial answer to the problem posed in \citet{Jackson2023SPA}. A comprehensive study on the bounded solution of one-dimensional quadratic BSDEs with unbounded stochastic parameters is provided for deriving our main results.