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.

Keywords:

• Backward stochastic differential equation
• existence and uniqueness
• comparison theorem
• g-supermartingale
• nonlinear Doob-Meyer’s decomposition

AMS Subject Classifications:

• 60H10
• 60H30

Acknowledgements

The authors would like to express great thanks to the anonymous referee for his/her careful reading and helpful suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11371362], [grant number 11601509]; the Natural Science Foundation of Jiangsu Province [grant number BK20150167]; the Research Innovation Program for College Graduates of Jiangsu Province [grant number KYZZ15_0376].