Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909089559216128 |
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| author | Hao, Tao Hu, Ying Tang, Shanjian Wen, Jiaqiang |
| author_facet | Hao, Tao Hu, Ying Tang, Shanjian Wen, Jiaqiang |
| contents | In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05676 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth Hao, Tao Hu, Ying Tang, Shanjian Wen, Jiaqiang Probability 60H10, 60H30 In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE. |
| title | Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth |
| topic | Probability 60H10, 60H30 |
| url | https://arxiv.org/abs/2211.05676 |