General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general \(g\)-supermartingales. (English) Zbl 1394.60066
Summary: 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Keywords:
backward stochastic differential equation; existence and uniqueness; comparison theorem; \(g\)-supermartingale; nonlinear Doob-Meyer’s decompositionReferences:
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