Viscosity solutions of nonlinear variational inequalities and their applications

非线性变分不等式的粘度解及其应用

• 批准号: 16540160
• 负责人: MORIMOTO Hiroaki
• 金额: $ 2.05万
• 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: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540160/
• 关键词: variational inequalities; viscosity solutions; Dynamic programming; nonlinear elliptic equations; 確率制御; 非線形楕円型方程式; Hamilton-Jacobi-Bellman方程式

The purpose of this study is to solve the optimization problems in mathematical economics and mathematical finance by applications of the modern theory in stochastic control. The study in this period focuses on the smoothness of viscosity solutions of the nonlinear variational inequalities and elliptic equations.Due to the grant, the results will be published in the book entitled:Stochastic Control and Mathematical Modelling in Economics.The main idea to solve these optimization problems is summarized as follows:(a) We formulate the problem and define the value function.(b) We verify that the Dynamic Programming Principle (DPP) holds for the value function.(c) By the DPP, the value function becomes a unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the problem.(d) By the uniqueness of viscosity solutions and the existence of a unique classical solution of the boundary value problem for the HJB equation, we obtain the smoothness of the viscosity solution of the HJB equation.(e) By the solution of the HJB equation, we construct an optimal policy.

这项研究的目的是通过随机控制中的现代理论的应用来解决数学经济学和数学融资中的优化问题。此期间的研究重点是非线性变化不平等和椭圆方程的粘度解决方案的平稳性。优化问题总结如下：（a）我们提出问题并定义了值函数。（b）我们验证动态编程原理（DPP）是否为该值函数保留。（c）由DPP，值函数变为汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程的独特粘度解决方案与问题相关。（d）通过粘度解决方案的唯一性以及HJB方程边界值问题的独特经典解决方案的存在，我们获得了HJB方程的粘度解的平滑度。（e）通过HJB方程的解，我们构建了一个最佳策略。

项目成果

Long-run average welfarte in a pollusion accumlation model

污染累积模型中的长期平均福利

• 发表时间: 2007
• 期刊: Journal of Economic Dynamic and Control 31
• 作者: K.Kawaguchi;H.Morimoto
• 通讯作者: H.Morimoto

Long-run average welfare in a pollution accumulation model

污染累积模型中的长期平均福利

• 发表时间: 2007
• 期刊: Journal of Economic Dynamics and Control 31
• 作者: K.Kawaguchi;H.Morimoto
• 通讯作者: H.Morimoto

経済学はじめの1歩-モデル分析の基礎

经济学的第一步——模型分析基础

• 发表时间: 2005
• 作者: 川口和仁;森本宏明
• 通讯作者: 森本宏明

The probabilistic solution of the Dirichlet problem for degenerate elliptic equations

简并椭圆方程狄利克雷问题的概率解

• 发表时间: 2006
• 期刊: Bulletin of the Polish Academy of Sciences 54
• 作者: Morimoto;Hiroaki
• 通讯作者: Hiroaki

An extension of the linear regulator for degenerate diffusions

简并扩散线性调节器的扩展

• 发表时间: 2005
• 期刊: IEEE Transactions on Automatic Control 50
• 作者: Md.A.Baten;H.Morimoto
• 通讯作者: H.Morimoto