Theory of viscosity solutions for nonlinear variational inequalities and its applications

• Project/Area Number: 21540188
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ehime University
• Principal Investigator: MORIMOTO Hiroaki 愛媛大学, 大学院・理工学研究科, 教授 (40085494)
• Co-Investigator(Kenkyū-buntansha): KAWAGUCHI Kazuhito 愛媛大学, 法文学部, 准教授 (30234040) ; ISHIKAWA Yasushi 愛媛大学, 大学院・理工学研究科, 准教授 (70202976)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 変分不等式 / 粘性解 / penalty法 / 幾何ブラウン運動 / 確率制御

Research Abstract

The study explored the optimization problem to maximize the utility function for the state described by the stochastic differential equations and the problem of nonlinear variational inequalities on mathematical economics. The research results can be summarized as follows:
(1) Publication of a book
This book provides the mathematical foundation for broader applications to mathematical economics.
(2) Development of the penalty method
The penalty equations are developed to solve nonlinear variational inequalities with gradient constraint.

Research Products

• [Journal Article] A singular control problem with discretionary stopping for geometric Brownian motions (2010)
  • Author(s): H. Morimoto
  • Journal Title: SIAM J. Control Optim Volume: 48 Pages: 3781-3804
• [Journal Article] Optimal dividend payments in the stochastic Ramsey model (2010)
  • Author(s): H. Morimoto
  • Journal Title: Stochastic Process. Appl Volume: 120 Pages: 427-441
• [Journal Article] Optimal dividend payments in the stochastic Ramsey model (2010)
  • Author(s): H.Morimoto
  • Journal Title: Stochastic Process.Appl. 120 Pages: 427-441
  • Peer Reviewed
• [Journal Article] A singular control problem with discretionary stopping for geometric Brownian motions (2010)
  • Author(s): H.Morimoto
  • Journal Title: SIAM J.Control and Optim. 48 Pages: 3781-3804
  • Peer Reviewed
• [Journal Article] Optimal consumption of the finite time horizon Ramsey problem (2009)
  • Author(s): T. Adachi, H. Morimoto
  • Journal Title: J. Math. Anal. Appl Volume: 358 Pages: 28-46
• [Journal Article] Optimal consumption of the finite time horizon Ramsey problem (2009)
  • Author(s): T.Adachi, H.Morimoto
  • Journal Title: J.Math.Anal.Appl. 358 Pages: 28-46
  • Peer Reviewed
• [Journal Article] An irreversible investment problem with maintenance expenditure
  • Author(s): K. Kawaguchi, H. Morimoto
  • Journal Title: SIAM J. Control Optim Volume: (印刷中)
• [Journal Article] Optimal dividend and risk control in diffusion models with linear costs
  • Author(s): H. Morimoto
  • Journal Title: OR2011 Proceedings Volume: (掲載予定)
• [Presentation] Variational inequalities in the optimal dividend and risk control problem (2012)
  • Organizer: 環瀬戸内ワークショッップ
  • Place of Presentation: 愛媛大学
• [Presentation] Optimal dividend and risk control in diffusion models with linear costs (2011)
  • Author(s): Hiroaki Morimoto
  • Organizer: OR 2011 Zurich
  • Place of Presentation: University of Zurich, Switzerland
  • Year and Date: 2011-08-31
• [Presentation] Parabolic variational inequalities for optimal dividends in the logistic model (2011)
  • Author(s): 森本宏明
  • Organizer: 松山解析セミナー2011
  • Place of Presentation: 愛媛大学理学部
  • Year and Date: 2011-02-04
• [Presentation] Optimal dividend and risk control in diffusion models with linear costs (2011)
  • Organizer: OR2011 Zurich
  • Place of Presentation: University of Zurich
• [Presentation] Parabolic variational inequalities for optimal dividends in the logistic model (2011)
  • Organizer: 松山解析セミナー
  • Place of Presentation: 愛媛大学
• [Presentation] On the penalty method in an irreversible investment problem (2010)
  • Author(s): 森本宏明
  • Organizer: 微分方程式深江ワークショップ
  • Place of Presentation: 神戸大学海事科学研究科
  • Year and Date: 2010-01-22
• [Presentation] On the penalty method in an irreversible investment problem (2010)
  • Organizer: 微分方程式深江ワークショップ
  • Place of Presentation: 神戸大学
• [Presentation] 幾何ブラウン運動に対する singular control と optimal stopping (2009)
  • Author(s): 森本宏明
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 愛媛大学メディアホール
  • Year and Date: 2009-12-15
• [Presentation] Variational inequalities with gradient constraint and applications to optimal dividend payments (2009)
  • Author(s): 森本宏明
  • Organizer: 微分方程式の粘性解とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-26
• [Presentation] 幾何ブラウン運動に対するsingular controlとoptimal stopping (2009)
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 愛媛大学
• [Presentation] Variational inequalities with gradient constraint and applications to optimal dividend payments (2009)
  • Organizer: 微分方程式の粘性解とその周辺
  • Place of Presentation: 京都大学数理研
• [Book] Cambridge University Press (2010)
  • Author(s): H. Morimoto
  • Total Pages: 325
  • Publisher: Stochastic Control and Mathematical Modeling : Applications in Economics
• [Book] Stochastic Control and Mathematical Modeling : Applications in Economics (2010)
  • Author(s): Hiroaki Morimoto
  • Total Pages: 325
  • Publisher: Cambridge University Press