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Nonlinear elliptic and parabolic PDEs, theories and applications

非线性椭圆和抛物线偏微分方程、理论和应用

基本信息

批准号：
14540148
负责人：
ARISAWA Mariko
金额：
$ 2.62万
依托单位：
Tohoku 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

项目摘要

This project was motivated to develop new methods, explore new problems and applications in the field of nonlinear elliptic and parabolic PDEs. In particular, we are interested in those problems arising in optimal controls of stochastic processes and deterministic dynamical systems.In the first year, a treatment of the stochastic PDEs by the viscosity solutions approach was studied, and at the same time some asymptotic problem arising in mathematical finances were begun to be considered. The head investigator profited six months on leave at the university of Paris 9 for some collaborations on the above subjects (Inamori fund). Moreover, some numerical experiments are tried to those problems above with collaborators. In the second year, some problems in mathematical finances are studied : e.g. the optimal portfolio construction with transaction costs, and the option pricing problems for the stock process containing jump terms. The former leads a free boundary problem (including some asymptotic problems), and the latter leads a class of integro-partial differential equations. These problems are significant from theoretical and practical point views. The head investigator wrote a paper on the regularity of degenerate elliptic equations, concerning with stochastic control, and she also prepares a work on the nonlinear first order PDE with or without shocks, concerning with the problem of the stochastic PDE. The mathematical finances work written above is also in preparation.In addition to the above, the head investigator have taught a course in GSIS, Tohoku University on nonlinear elliptic and parabolic PDEs, optimal control problems arising in mathematical finances, and the basic theory of numerical analysis.

本项目的目的是在非线性椭圆和抛物偏微分方程领域发展新的方法，探索新的问题和应用。特别是，我们感兴趣的随机过程和确定性动力系统的最优控制中出现的那些问题。在第一年，用粘性解方法研究了随机偏微分方程的处理，同时开始考虑数学金融中出现的一些渐近问题。首席研究员在巴黎9大学休假六个月，就上述主题进行了一些合作（稻盛基金）。此外，本文还与合作者对上述问题进行了数值试验。第二年，研究了金融数学中的一些问题，如有交易费用的最优投资组合的构造问题，以及含跳跃项的股票过程的期权定价问题。前者导出一个自由边界问题（包括一些渐近问题），后者导出一类积分-偏微分方程。这些问题具有重要的理论意义和现实意义。首席研究员写了一篇关于退化椭圆方程的正则性的论文，涉及随机控制，她还准备了一项关于有或无冲击的非线性一阶PDE的工作，涉及随机PDE的问题。此外，还在东北大学GSIS教授了非线性椭圆型和抛物型偏微分方程、金融数学中的最优控制问题、数值分析的基本理论等课程。