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.

该项目旨在开发新方法，探索非线性椭圆和抛物线偏微分方程领域的新问题和应用。我们特别对随机过程和确定性动力系统的最优控制中出现的问题感兴趣。第一年，研究了用粘度解方法处理随机偏微分方程，同时研究了随机过程中出现的一些渐近问题。人们开始考虑数学金融。首席研究员利用巴黎第九大学六个月的休假来进行上述课题的一些合作（稻盛基金）。此外，针对上述问题，与合作者进行了一些数值实验。第二年，研究数学金融中的一些问题：例如具有交易成本的最优投资组合构建，以及包含跳跃项的股票过程的期权定价问题。前者导致自由边界问题（包括一些渐近问题），后者导致一类积分偏微分方程。这些问题从理论和实践的角度来看都很重要。首席研究员写了一篇关于简并椭圆方程正则性的论文，涉及随机控制，她还准备了一篇关于带或不带冲击的非线性一阶偏微分方程的工作，涉及随机偏微分方程的问题。上面写的数学金融工作也在准备中。除上述之外，首席研究员还在东北大学 GSIS 讲授过非线性椭圆和抛物线偏微分方程、数学金融中出现的最优控制问题以及数值数学基础理论等课程。分析。