A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916173759643648 |
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| author | Burzoni, Matteo Cecchin, Alekos Cosso, Andrea |
| author_facet | Burzoni, Matteo Cecchin, Alekos Cosso, Andrea |
| contents | We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean-Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the ''fast'' variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose ''fast'' component converges to the optimal control process, while the ''slow'' component converges to the optimal state process. The interest in such a procedure is that it allows to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12293 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems Burzoni, Matteo Cecchin, Alekos Cosso, Andrea Probability Optimization and Control 60H10, 49N80, 49M99 We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean-Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the ''fast'' variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose ''fast'' component converges to the optimal control process, while the ''slow'' component converges to the optimal state process. The interest in such a procedure is that it allows to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian. |
| title | A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems |
| topic | Probability Optimization and Control 60H10, 49N80, 49M99 |
| url | https://arxiv.org/abs/2212.12293 |