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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/2406.00286 |
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| _version_ | 1866929369307414528 |
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| author | Li, Juan Li, Zhanxin Xing, Chuanzhi |
| author_facet | Li, Juan Li, Zhanxin Xing, Chuanzhi |
| contents | For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of $Z$-component of the solution process $(Y, Z)$. A natural question is whether general mean-field BSDEs whose coefficients depend on the law of $Z$ have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process $(Y,Z)$. With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component $Y$ of the solution $(Y,Z)$ for such mean-field BSDEs, but also the second component $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution $(Y,Z)$ Li, Juan Li, Zhanxin Xing, Chuanzhi Probability For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of $Z$-component of the solution process $(Y, Z)$. A natural question is whether general mean-field BSDEs whose coefficients depend on the law of $Z$ have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process $(Y,Z)$. With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component $Y$ of the solution $(Y,Z)$ for such mean-field BSDEs, but also the second component $Z$. |
| title | Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution $(Y,Z)$ |
| topic | Probability |
| url | https://arxiv.org/abs/2406.00286 |