Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. (English) Zbl 1333.60120
Summary: In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose \(i\)-th component depends only on the \(i\)-th row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in [N. El-Karoui and S. Hamadène, Stochastic Processes Appl. 107, No. 1, 145–169 (2003; Zbl 1075.60534), p.164].
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 93E20 | Optimal stochastic control |
Keywords:
muliti-dimensional backward stochastic differential equations; diagonally quadratic generators; BMO martingales; John-Nirenberg inequality; reverse Hölder inequalityCitations:
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