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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.17560 |
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| _version_ | 1866917630592417792 |
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| author | Wang, Yan Li, Xinying Gu, Chuang Fan, Shengjun |
| author_facet | Wang, Yan Li, Xinying Gu, Chuang Fan, Shengjun |
| contents | With the terminal value $ξ^-$ admitting a certain exponential moment and $ξ^+$ admitting every exponential moments or being bounded, we establish several existence and uniqueness results for unbounded solutions of backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a monotonicity condition with general growth in the first unknown variable $y$ and a convexity condition with quadratic growth in the second unknown variable $z$. In particular, the generator $g$ may be not locally-Lipschitz continuous in $y$. This generalizes some results reported in \cite{Delbaen 2011} by relaxing the continuity and growth of $g$ in $y$. We also give an explicit expression of the first process in the unique unbounded solution of a BSDE when the generator $g$ is jointly convex in $(y,z)$ and has a linear growth in $y$ and a quadratic growth in $z$. Finally, we put forward the corresponding comparison theorems for unbounded solutions of the preceding BSDEs. These results are proved by those existing ideas and some innovative ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence and uniqueness of unbounded solutions to quadratic BSDEs with monotonic-convex generators Wang, Yan Li, Xinying Gu, Chuang Fan, Shengjun Probability With the terminal value $ξ^-$ admitting a certain exponential moment and $ξ^+$ admitting every exponential moments or being bounded, we establish several existence and uniqueness results for unbounded solutions of backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a monotonicity condition with general growth in the first unknown variable $y$ and a convexity condition with quadratic growth in the second unknown variable $z$. In particular, the generator $g$ may be not locally-Lipschitz continuous in $y$. This generalizes some results reported in \cite{Delbaen 2011} by relaxing the continuity and growth of $g$ in $y$. We also give an explicit expression of the first process in the unique unbounded solution of a BSDE when the generator $g$ is jointly convex in $(y,z)$ and has a linear growth in $y$ and a quadratic growth in $z$. Finally, we put forward the corresponding comparison theorems for unbounded solutions of the preceding BSDEs. These results are proved by those existing ideas and some innovative ones. |
| title | On the existence and uniqueness of unbounded solutions to quadratic BSDEs with monotonic-convex generators |
| topic | Probability |
| url | https://arxiv.org/abs/2401.17560 |