Brownian particles controlled by their occupation measure
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916695047667712 |
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| author | Béthencourt, Loïc Catellier, Rémi Tanré, Etienne |
| author_facet | Béthencourt, Loïc Catellier, Rémi Tanré, Etienne |
| contents | In this article, we study a finite horizon linear-quadratic stochastic control problem for Brownian particles, where the cost functions depend on the state and the occupation measure of the particles. To address this problem, we develop an Itô formula for the flow of occupation measure, which enables us to derive the associated Hamilton-Jacobi-Bellman equation. Then, thanks to a Feynman-Kac formula and the Boué-Dupuis formula, we construct an optimal strategy and an optimal trajectory. Finally, we illustrate our result when the cost-function is the volume of the sausage associated to the particles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brownian particles controlled by their occupation measure Béthencourt, Loïc Catellier, Rémi Tanré, Etienne Probability 93E20, 49J55, 60G57, 49L12 In this article, we study a finite horizon linear-quadratic stochastic control problem for Brownian particles, where the cost functions depend on the state and the occupation measure of the particles. To address this problem, we develop an Itô formula for the flow of occupation measure, which enables us to derive the associated Hamilton-Jacobi-Bellman equation. Then, thanks to a Feynman-Kac formula and the Boué-Dupuis formula, we construct an optimal strategy and an optimal trajectory. Finally, we illustrate our result when the cost-function is the volume of the sausage associated to the particles. |
| title | Brownian particles controlled by their occupation measure |
| topic | Probability 93E20, 49J55, 60G57, 49L12 |
| url | https://arxiv.org/abs/2404.06960 |