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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.13450 |
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| _version_ | 1866917813537472512 |
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| author | Li, Shang |
| author_facet | Li, Shang |
| contents | We derive an Itô-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of Itô's formula. To address the new challenges arising from the measure-valued setting, we employ symmetric polynomials to approximate the second-order linear derivative of the functional on finite measures, alongside certain localization techniques.
A controlled superprocess with a binary branching mechanism can be interpreted as a weak solution to a controlled stochastic partial differential equation (SPDE), which naturally leads to such a decomposition. Consequently, this Itô-type formula makes it possible to derive the Hamilton-Jacobi-Bellman (HJB) equation and the verification theorem for controlled superprocesses with a binary branching mechanism. Additionally, we propose a heuristic definition for the viscosity solution of an equation involving derivatives on finite measures. We prove that a continuous value function is a viscosity solution in this sense and demonstrate the uniqueness of the viscosity solution when the second-order derivative term on the measure vanishes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Itô-type formula for some measure-valued processes and its application on controlled superprocesses Li, Shang Probability Optimization and Control We derive an Itô-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of Itô's formula. To address the new challenges arising from the measure-valued setting, we employ symmetric polynomials to approximate the second-order linear derivative of the functional on finite measures, alongside certain localization techniques. A controlled superprocess with a binary branching mechanism can be interpreted as a weak solution to a controlled stochastic partial differential equation (SPDE), which naturally leads to such a decomposition. Consequently, this Itô-type formula makes it possible to derive the Hamilton-Jacobi-Bellman (HJB) equation and the verification theorem for controlled superprocesses with a binary branching mechanism. Additionally, we propose a heuristic definition for the viscosity solution of an equation involving derivatives on finite measures. We prove that a continuous value function is a viscosity solution in this sense and demonstrate the uniqueness of the viscosity solution when the second-order derivative term on the measure vanishes. |
| title | An Itô-type formula for some measure-valued processes and its application on controlled superprocesses |
| topic | Probability Optimization and Control |
| url | https://arxiv.org/abs/2410.13450 |