On the Singular Control of a Diffusion and Its Running Infimum or Supremum
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| Format: | Preprint |
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2025
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| _version_ | 1866915127818715136 |
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| author | Ferrari, Giorgio Rodosthenous, Neofytos |
| author_facet | Ferrari, Giorgio Rodosthenous, Neofytos |
| contents | We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_17577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Singular Control of a Diffusion and Its Running Infimum or Supremum Ferrari, Giorgio Rodosthenous, Neofytos Optimization and Control Probability Mathematical Finance 93E20, 60J60, 49L12, 91B70 We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance. |
| title | On the Singular Control of a Diffusion and Its Running Infimum or Supremum |
| topic | Optimization and Control Probability Mathematical Finance 93E20, 60J60, 49L12, 91B70 |
| url | https://arxiv.org/abs/2501.17577 |