Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions
Abstract A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is ba...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-09-01
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| Series: | Journal of Mathematics in Industry |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13362-021-00112-9 |
| _version_ | 1818935034760921088 |
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| author | Hidekazu Yoshioka Yuta Yaegashi |
| author_facet | Hidekazu Yoshioka Yuta Yaegashi |
| author_sort | Hidekazu Yoshioka |
| collection | DOAJ |
| description | Abstract A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is based on a dynamic programming formalism to impulsively control a 1-D diffusion process of a geometric Brownian type. The imperfectness leads to a non-local operator different from the many conventional ones, and evokes a slightly different optimal intervention policy. We give viscosity characterizations of the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) governing the value function focusing on its numerical computation. Uniqueness and verification results of the HJBQVI are presented and a candidate exact solution is constructed. The HJBQVI is solved with the two different numerical methods, an ordinary differential equation (ODE) based method and a finite difference scheme, demonstrating their consistency. Furthermore, the resulting controlled dynamics are extensively analyzed focusing on a bird population management case from a statistical standpoint. |
| first_indexed | 2024-12-20T05:13:45Z |
| format | Article |
| id | doaj.art-b4634c46a9e744548b07a851f032cc60 |
| institution | Directory Open Access Journal |
| issn | 2190-5983 |
| language | English |
| last_indexed | 2024-12-20T05:13:45Z |
| publishDate | 2021-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Mathematics in Industry |
| spelling | doaj.art-b4634c46a9e744548b07a851f032cc602022-12-21T19:52:13ZengSpringerOpenJournal of Mathematics in Industry2190-59832021-09-0111113410.1186/s13362-021-00112-9Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventionsHidekazu Yoshioka0Yuta Yaegashi1Assistant Professor, Graduate School of Natural Science and Technology, Shimane UniversityIndependent Researcher, Dr. of Agr.Abstract A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is based on a dynamic programming formalism to impulsively control a 1-D diffusion process of a geometric Brownian type. The imperfectness leads to a non-local operator different from the many conventional ones, and evokes a slightly different optimal intervention policy. We give viscosity characterizations of the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) governing the value function focusing on its numerical computation. Uniqueness and verification results of the HJBQVI are presented and a candidate exact solution is constructed. The HJBQVI is solved with the two different numerical methods, an ordinary differential equation (ODE) based method and a finite difference scheme, demonstrating their consistency. Furthermore, the resulting controlled dynamics are extensively analyzed focusing on a bird population management case from a statistical standpoint.https://doi.org/10.1186/s13362-021-00112-9Imperfect impulse controlPopulation dynamicsHamilton–Jacobi–Bellman quasi variational inequalityODE-based methodFinite difference scheme |
| spellingShingle | Hidekazu Yoshioka Yuta Yaegashi Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions Journal of Mathematics in Industry Imperfect impulse control Population dynamics Hamilton–Jacobi–Bellman quasi variational inequality ODE-based method Finite difference scheme |
| title | Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| title_full | Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| title_fullStr | Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| title_full_unstemmed | Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| title_short | Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| title_sort | mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions |
| topic | Imperfect impulse control Population dynamics Hamilton–Jacobi–Bellman quasi variational inequality ODE-based method Finite difference scheme |
| url | https://doi.org/10.1186/s13362-021-00112-9 |
| work_keys_str_mv | AT hidekazuyoshioka mathematicalandnumericalanalysesofastochasticimpulsecontrolmodelwithimperfectinterventions AT yutayaegashi mathematicalandnumericalanalysesofastochasticimpulsecontrolmodelwithimperfectinterventions |