Integral equations for the optimal boundary surface of a mean-field game of capacity expansion
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| Format: | Preprint |
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2026
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| _version_ | 1866915936012861440 |
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| author | De Angelis, Tiziano Livieri, Giulia Ghio, Maddalena |
| author_facet | De Angelis, Tiziano Livieri, Giulia Ghio, Maddalena |
| contents | We prove that the optimal boundary surface that splits the action and inaction regions in a mean-field game of capacity expansion studied in (Campi et al.,\ Ann.\ Appl.\ Probab.,\ {\bf 32}(5),\, pp.\,3674-3717, 2022) is the unique continuous solution of a nonlinear integral equation of Volterra type. In order to do that, we first establish continuity of the optimal surface. Then we develop an extension of Itô's formula which weakens assumptions required in the existing literature on the first-order time-derivative and/or second-order space derivative of the value function. The paper also provides an algorithm for the numerical solution of the integral equation and we compute optimal controls numerically for the mean-field game. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12061 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral equations for the optimal boundary surface of a mean-field game of capacity expansion De Angelis, Tiziano Livieri, Giulia Ghio, Maddalena Optimization and Control 91A16, 93E20, 60G40, 45D05, 65R20 We prove that the optimal boundary surface that splits the action and inaction regions in a mean-field game of capacity expansion studied in (Campi et al.,\ Ann.\ Appl.\ Probab.,\ {\bf 32}(5),\, pp.\,3674-3717, 2022) is the unique continuous solution of a nonlinear integral equation of Volterra type. In order to do that, we first establish continuity of the optimal surface. Then we develop an extension of Itô's formula which weakens assumptions required in the existing literature on the first-order time-derivative and/or second-order space derivative of the value function. The paper also provides an algorithm for the numerical solution of the integral equation and we compute optimal controls numerically for the mean-field game. |
| title | Integral equations for the optimal boundary surface of a mean-field game of capacity expansion |
| topic | Optimization and Control 91A16, 93E20, 60G40, 45D05, 65R20 |
| url | https://arxiv.org/abs/2604.12061 |