General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

TL;DR

This paper proves existence and uniqueness of solutions for multidimensional BSDEs with generators under weak stochastic-monotonicity and growth conditions, extending previous results and introducing new inequalities.

Contribution

It introduces generalized stochastic Gronwall and Bihari inequalities and applies them to establish well-posedness of multidimensional BSDEs with relaxed conditions.

Findings

Established existence and uniqueness of solutions for the class of BSDEs.

Developed stochastic Gronwall and Bihari inequalities.

Unified and strengthened previous results on multidimensional BSDEs.

Abstract

This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other places. The martingale representation theorem, It\^{o}'s formula and the BMO martingale tool are used to prove these two inequalities.