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

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 \begin{document}$ g $\end{document} satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable \begin{document}$ y $\end{document} , and a stochastic-Lipschitz condition in the state variable \begin{document}$ z $\end{document} . This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. 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 applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.