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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/2410.01543 |
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| _version_ | 1866916420730748928 |
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| author | Li, Xinying Fan, Shengjun |
| author_facet | Li, Xinying Fan, Shengjun |
| contents | In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $τ$, which may take values in $[0,+\infty]$. Firstly, we establish an existence and uniqueness result for a weighted $L^p~(p>1)$ solution of the preceding BSDE with generator $g$ satisfying a stochastic monotonicity condition with general growth in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$. Then, we derive an existence and uniqueness result for a weighted $L^1$ solution of the preceding BSDE under an additional stochastic sub-linear growth condition in $z$. These results generalize the corresponding ones obtained in \cite{Li2024} to the $L^p~(p\geq 1)$ solution case. Finally, the corresponding comparison theorems for the weighted $L^p~(p\geq1)$ solutions are also put forward and verified in the one-dimensional setting. In particular, we develop new ideas and systematical techniques in order to establish the above results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01543 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted $L^p~(p\geq1)$ solutions of random time horizon BSDEs with stochastic monotonicity generators Li, Xinying Fan, Shengjun Probability In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $τ$, which may take values in $[0,+\infty]$. Firstly, we establish an existence and uniqueness result for a weighted $L^p~(p>1)$ solution of the preceding BSDE with generator $g$ satisfying a stochastic monotonicity condition with general growth in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$. Then, we derive an existence and uniqueness result for a weighted $L^1$ solution of the preceding BSDE under an additional stochastic sub-linear growth condition in $z$. These results generalize the corresponding ones obtained in \cite{Li2024} to the $L^p~(p\geq 1)$ solution case. Finally, the corresponding comparison theorems for the weighted $L^p~(p\geq1)$ solutions are also put forward and verified in the one-dimensional setting. In particular, we develop new ideas and systematical techniques in order to establish the above results. |
| title | Weighted $L^p~(p\geq1)$ solutions of random time horizon BSDEs with stochastic monotonicity generators |
| topic | Probability |
| url | https://arxiv.org/abs/2410.01543 |