Controlled superprocesses and HJB equation in the space of finite measures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913579399118848 |
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| author | Ocello, Antonio |
| author_facet | Ocello, Antonio |
| contents | This paper introduces the formalism required to analyze a certain class of stochastic control problems that involve a super diffusion as the underlying controlled system. To establish the existence of these processes, we show that they are weak scaling limits of controlled branching processes. First, we prove a generalized Itô's formula for this dynamics in the space of finite measures, using the differentiation in the space of finite positive measures. This lays the groundwork for a PDE characterization of the value function of a control problem, which leads to a verification theorem. Finally, focusing on an exponential-type value function, we show how a regular solution to a finite--dimensional HJB equation can be used to construct a smooth solution to the HJB equation in the space of finite measures, via the so-called branching property technique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15962 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Controlled superprocesses and HJB equation in the space of finite measures Ocello, Antonio Probability 93E20, 49L20, 49L12, 60J70, 35K10, 35Q93, 60H30 This paper introduces the formalism required to analyze a certain class of stochastic control problems that involve a super diffusion as the underlying controlled system. To establish the existence of these processes, we show that they are weak scaling limits of controlled branching processes. First, we prove a generalized Itô's formula for this dynamics in the space of finite measures, using the differentiation in the space of finite positive measures. This lays the groundwork for a PDE characterization of the value function of a control problem, which leads to a verification theorem. Finally, focusing on an exponential-type value function, we show how a regular solution to a finite--dimensional HJB equation can be used to construct a smooth solution to the HJB equation in the space of finite measures, via the so-called branching property technique. |
| title | Controlled superprocesses and HJB equation in the space of finite measures |
| topic | Probability 93E20, 49L20, 49L12, 60J70, 35K10, 35Q93, 60H30 |
| url | https://arxiv.org/abs/2306.15962 |