Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913634126397440 |
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| author | Bayraktar, Erhan Cheung, Hang Ekren, Ibrahim Qiu, Jinniao Tai, Ho Man Zhang, Xin |
| author_facet | Bayraktar, Erhan Cheung, Hang Ekren, Ibrahim Qiu, Jinniao Tai, Ho Man Zhang, Xin |
| contents | In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space Bayraktar, Erhan Cheung, Hang Ekren, Ibrahim Qiu, Jinniao Tai, Ho Man Zhang, Xin Optimization and Control Analysis of PDEs Probability 49L25, 35Q93, 35B51, 58E30 In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature. |
| title | Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space |
| topic | Optimization and Control Analysis of PDEs Probability 49L25, 35Q93, 35B51, 58E30 |
| url | https://arxiv.org/abs/2501.01612 |