Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914487074816000 |
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| author | Zheng, Shiqiu |
| author_facet | Zheng, Shiqiu |
| contents | In this paper, we investigate the well-posedness of bounded and unbounded solutions for reflected backward stochastic differential equations (RBSDEs) and backward stochastic differential equations (BSDEs). The generators of these equations satisfy a one-sided growth restriction on the variable $y$ and have a general quadratic growth in the variable $z$. The solutions $Y_t$ (and the obstacles for RBSDEs) take values in either $\mathbf{R}$ or $(0, \infty)$. We obtain the existence of solutions primarily by using the methods from Essaky and Hassani (2011) and Bahlali et al. (2017). For the uniqueness of solutions, we provide a method applicable when the generators are convex in $(y,z)$ or are (locally) Lipschitz in $y$ and convex in $z$. Our method relies on the $θ$-difference technique introduced by Briand and Hu (2008), and some novel comparison arguments based on RBSDEs. We also establish some general comparison theorems for such RBSDEs and BSDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21172 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions Zheng, Shiqiu Probability In this paper, we investigate the well-posedness of bounded and unbounded solutions for reflected backward stochastic differential equations (RBSDEs) and backward stochastic differential equations (BSDEs). The generators of these equations satisfy a one-sided growth restriction on the variable $y$ and have a general quadratic growth in the variable $z$. The solutions $Y_t$ (and the obstacles for RBSDEs) take values in either $\mathbf{R}$ or $(0, \infty)$. We obtain the existence of solutions primarily by using the methods from Essaky and Hassani (2011) and Bahlali et al. (2017). For the uniqueness of solutions, we provide a method applicable when the generators are convex in $(y,z)$ or are (locally) Lipschitz in $y$ and convex in $z$. Our method relies on the $θ$-difference technique introduced by Briand and Hu (2008), and some novel comparison arguments based on RBSDEs. We also establish some general comparison theorems for such RBSDEs and BSDEs. |
| title | Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions |
| topic | Probability |
| url | https://arxiv.org/abs/2412.21172 |