Studies on non-linear variational inequalities by viscosity solutions

粘性解非线性变分不等式的研究

• 批准号: 14540178
• 负责人: MORIMOTO Hiroaki
• 金额: $ 1.41万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540178/
• 关键词: Variational inequality; viscosity solution; stochastic control; 確立制御; 非線形変分不等式; 確率微分方程式; 最適化

The purpose of this study is to solve optimization problems in mathematical economics and mathematical finance by applications of the modern theory in stochastic control The main result in this term lies in opening up a field of research of non-linear variational inequalities for the problem so as to minimize the cost functional with discretionary stopping. The content is as follows.(1)In the paper "Variational inequalities for combined control and stopping, 2003", the new definition of viscosity solutions for non-linear variational inequalities is given. The penalty method for linear variational inequalities originated by Bensoussan-Lions is developed, and it is shown that there exists a unique viscosity solution of the non-linear variational inequality. The game problem in the same setting is discussed.(2) In the paper "Variational inequalities for leavable bounded-velocity control", the key for the smoothness problem of the viscosity solutions to non-linear variational inequalities is presented. Recently, it becomes well known when the viscosity solutions of Hamilton-Jacobi-Bellman equations are smooth for the construction of optimal policies. The paper "Optimal exploitation of renewable resources by the viscosity solution method" shows that the similar method is valid for the optimal consumption problem of renewable resources in mathematical economics. These results can be applicable for various optimization problems in future.

这项研究的目的是通过现代理论在随机控制中的应用来解决数学经济学和数学金融中的优化问题。本术语的主要结果在于开放问题的非线性变异不平等研究领域通过可支配的停止来最大程度地降低成本功能。内容如下。（1）在论文中“合并控制和停止的变异不平等，2003”，给出了非线性变异不等式的粘度解决方案的新定义。开发了源自​​皮革的狮子的线性变异不平等的惩罚方法，并表明存在非线性变异不平等的独特粘度解决方案。讨论了同一环境中的游戏问题。（2）在“可弹性有限速度控制的变化不平等”中，提出了粘度解决方案平滑度问题的关键，即非线性变异不平等。最近，当汉密尔顿 - 雅各比 - 贝尔曼方程的粘度解决方案对于最佳策略的构建平滑时，众所周知。该论文“通过粘度解决方案方法对可再生资源的最佳开发”表明，类似的方法对于数学经济学中可再生资源的最佳消耗问题有效。这些结果可能适用于将来的各种优化问题。

项目成果

Kamizono, K., Morimoto, H.: "On a variational inequality associated with a stopping game combined with a control"Stochastics and Stochastisc Rep.. 70. 99-123 (2002)

Kamizono, K.、Morimoto, H.：“关于与停止游戏和控制相结合的变分不等式”Stochastics and Stochastics Rep.. 70. 99-123 (2002)

S.Koike, H.Morimoto: "Variational inequalities for leavable bounded-velocity control"Appl.Math.Optim.. 48. 1-20 (2003)

S.Koike、H.Morimoto：“可左有界速度控制的变分不等式”Appl.Math.Optim.. 48. 1-20 (2003)

S.Koike, H.Morimoto: "On variational inequalities for leavable bounded-velocity control"Appl. Math. Optim.. (掲載予定).

S.Koike，H.Morimoto：“关于左有界速度控制的变分不等式”应用数学。

H.Morimoto: "Variational inequalities for comfined control and stopping"SIAM J. Control and Optim. (掲載予定).

H.Morimoto：“受限控制和停止的变分不等式”SIAM J. Control and Optim（待出版）。

H.Morimoto, K.Kawaguchi: "Optimal exploitation of renewable resources by the viscosity solution method"Stoch. Anal. Appl.. 20. 927-946 (2002)

H.Morimoto、K.Kawaguchi：“通过粘度解决方法优化可再生资源的开发”Stoch。