Parisian reflection strategies and dynamic optimization

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K05377
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kansai University
• Principal Investigator: YAMAZAKI Kazutoshi 関西大学, システム理工学部, 准教授 (50554937)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: レヴィー過程 / ファイナンス / 保険 / 在庫管理 / 確率制御

Outline of Final Research Achievements

Levy processes are used in various fields such as finance, insurance and queues. Their recent developments have enabled us to achieve realistic models, generalizing greatly the traditional Brownian motion models. Regarding their applications in optimal stochastic control, analytical results such as those on reflected processes can be directly used. This research focused on a version of stochastic control where observation times are given by Poisson arrival times and developed analysis on optimal solutions and theories needed to achieve these. By studying further the related results on Levy processes observed at Poisson arrival times, we solved the optimal dividend, inventory and stopping problems, establishing procedures and techniques for the derivation of optimal solutions.

Free Research Field

確率論、数理ファイナンス、保険数学

Academic Significance and Societal Importance of the Research Achievements

連続時間確率制御問題はこれまで連続観測モデルに焦点が当てられ、伊藤の公式や微分方程式論を駆使することによって、様々な理論的結果が得られてきた。しかしながら、現実には観測は離散的に行われ、連続的観測モデルがその近似を正確に行えられるかを調べることは非常に重要な課題であった。一方で離散観測モデルでは一般的には解析的手法の利用が難しく、数値的アプローチに限られてきた。本研究では観測がポアソン的である場合に焦点を当てることで、解析的アプローチを確立させた。