On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914549589868544 |
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| author | Lei, Qian Pun, Chi Seng |
| author_facet | Lei, Qian Pun, Chi Seng |
| contents | This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach's fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and naïve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01986 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type Lei, Qian Pun, Chi Seng Analysis of PDEs Optimization and Control Mathematical Finance 93E20, 35A01, 35A02, 35K10, 35Q93, 49L12 This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach's fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and naïve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable. |
| title | On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type |
| topic | Analysis of PDEs Optimization and Control Mathematical Finance 93E20, 35A01, 35A02, 35K10, 35Q93, 49L12 |
| url | https://arxiv.org/abs/2307.01986 |