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.

本研究的目的是应用现代随机控制理论来解决数理经济学和数理金融学中的最优化问题。本研究的主要结果是开拓了问题的非线性变分不等式的研究领域，从而使具有任意停止的费用泛函最小化。内容如下。(1)In 2003年发表的“变分不等式联合控制与停止”一文中，给出了非线性变分不等式粘性解的新定义。发展了Bensoussan-Lions提出的线性变分不等式的罚方法，证明了非线性变分不等式存在唯一粘性解.讨论了在相同条件下的博弈问题。(2)在“可离开有界速度控制的变分不等式”一文中，给出了非线性变分不等式粘性解的光滑性问题的关键。近年来，当Hamilton-Jacobi-Bellman方程的粘性解是光滑的时，最优策略的构造问题已成为人们所熟知的。文章“粘性解方法在可再生资源最优开采中的应用”表明，类似的方法对于数理经济学中的可再生资源最优消费问题是有效的。这些结果可应用于未来的各种优化问题。

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。