2006 Fiscal Year Final Research Report Summary
Viscosity solutions of nonlinear variational inequalities and their applications
| Project/Area Number |
16540160
|
|---|
| 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 EhimeUniversity, Graduate School of Science and Engineering, Professor, 理工学研究科, 教授 (80166438)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KAWAGUCHI Kazuhito Ehime University, Faculty of Law and Letters, Associate Professor, 法文学部, 助教授 (30234040)
ISHIKAWA Yasushi Ehime University, Graduate School of Science and Engineering, Associate Professor, 理工学研究科, 助教授 (70202976)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | variational inequalities / viscosity solutions / Dynamic programming / nonlinear elliptic equations |
|---|
| Research Abstract |
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.
|
