General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general g-supermartingales

Abstract

The main purpose of this paper is to prove an existence and uniqueness result for solutions of a multidimensional backward stochastic differential equation (BSDE) with a general time interval (including the deterministic and stochastic cases), where the generator g of the BSDE is weakly monotonic and of general growth in y, and Lipschitz continuous in z, both non-uniformly with respect to t. And, the corresponding comparison theorem for the solutions of one-dimensional BSDEs is provided. As applications, we establish a nonlinear Doob-Meyer’s decomposition theorem for general continuous g-supermartingales under an additional assumption of the generator g. Some new problems in our setting arise naturally and are well overcome. These results generalize and improve some known works.