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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2510.11067 |
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| _version_ | 1866917007498149888 |
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| author | Jiang, Weimin Li, Juan Shen, Yan |
| author_facet | Jiang, Weimin Li, Juan Shen, Yan |
| contents | This paper investigates $L^{1}$ solutions for mean-field backward stochastic differential equations (MFBSDEs) under different weak assumptions in both one-dimensional and multi-dimensional settings, whose generator $f(ω,t,y,z,μ)$ depends not only on the solution process $(Y,Z)$ but also on the law of $(Y,Z)$. In the one-dimensional case where $f$ depends on the law of $Y$, we show with the help of a test function method and a localization procedure that such type of equations with an integrable terminal condition admits an $L^{1}$ solution, when the generator $f(ω,t,y,z,μ)$ has a one-sided linear growth in $(y,μ)$, and an iterated-logarithmically sub-linear growth in $z$. Furthermore, by leveraging the additional extended monotonicity in $y$ and an iterated-logarithmically uniform continuity in $z$ of the generator $f(ω,t,y,z,μ)$ together with a strengthened nondecreasing condition in $μ$, we derive a comparison theorem for $L^{1}$ solutions, which immediately leads to the uniqueness of the $L^{1}$ solutions. Next, we establish the existence and the uniqueness of $L^{1}$ solutions for multi-dimensional mean-field BSDEs with integrable parameters in which the generator $f(ω,t,y,z,μ)$ depends on $μ=\mathbb{P}_{Y}$ and satisfies a one-sided Osgood condition as well as a general growth condition in $y$, a Lipschitz continuity as well as a sublinear growth condition in $z$, and a Lipschitz condition in $μ$. Finally, the solvability of $L^{1}$ solutions for general MFBSDEs is studied, where the generator $f(ω,t,y,z,μ)$ depends on both the solution process $(Y,Z)$ and its joint law $\mathbb{P}_{(Y,Z)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General mean-field BSDEs with integrable terminal values Jiang, Weimin Li, Juan Shen, Yan Probability This paper investigates $L^{1}$ solutions for mean-field backward stochastic differential equations (MFBSDEs) under different weak assumptions in both one-dimensional and multi-dimensional settings, whose generator $f(ω,t,y,z,μ)$ depends not only on the solution process $(Y,Z)$ but also on the law of $(Y,Z)$. In the one-dimensional case where $f$ depends on the law of $Y$, we show with the help of a test function method and a localization procedure that such type of equations with an integrable terminal condition admits an $L^{1}$ solution, when the generator $f(ω,t,y,z,μ)$ has a one-sided linear growth in $(y,μ)$, and an iterated-logarithmically sub-linear growth in $z$. Furthermore, by leveraging the additional extended monotonicity in $y$ and an iterated-logarithmically uniform continuity in $z$ of the generator $f(ω,t,y,z,μ)$ together with a strengthened nondecreasing condition in $μ$, we derive a comparison theorem for $L^{1}$ solutions, which immediately leads to the uniqueness of the $L^{1}$ solutions. Next, we establish the existence and the uniqueness of $L^{1}$ solutions for multi-dimensional mean-field BSDEs with integrable parameters in which the generator $f(ω,t,y,z,μ)$ depends on $μ=\mathbb{P}_{Y}$ and satisfies a one-sided Osgood condition as well as a general growth condition in $y$, a Lipschitz continuity as well as a sublinear growth condition in $z$, and a Lipschitz condition in $μ$. Finally, the solvability of $L^{1}$ solutions for general MFBSDEs is studied, where the generator $f(ω,t,y,z,μ)$ depends on both the solution process $(Y,Z)$ and its joint law $\mathbb{P}_{(Y,Z)}$. |
| title | General mean-field BSDEs with integrable terminal values |
| topic | Probability |
| url | https://arxiv.org/abs/2510.11067 |