Mean-field control of non exchangeable systems
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909269598666752 |
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| author | De Crescenzo, Anna Fuhrman, Marco Kharroubi, Idris Pham, Huyên |
| author_facet | De Crescenzo, Anna Fuhrman, Marco Kharroubi, Idris Pham, Huyên |
| contents | We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a $L^2$-set of Wasserstein space-valued functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean-field control of non exchangeable systems De Crescenzo, Anna Fuhrman, Marco Kharroubi, Idris Pham, Huyên Probability Optimization and Control 60H30, 05C80, 60K35, 93E20 We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a $L^2$-set of Wasserstein space-valued functions. |
| title | Mean-field control of non exchangeable systems |
| topic | Probability Optimization and Control 60H30, 05C80, 60K35, 93E20 |
| url | https://arxiv.org/abs/2407.18635 |