| Co-Investigator(Kenkyū-buntansha) |
URAKAWA Hajime Tohoku University, GSIS, Prof., 大学院・情報科学研究科, 教授 (50022679)
GIGA Yoshikazu Hokkaido University, Dept.of Maths., Prof., 大学院・理学研究科, 教授 (70144110)
MIKAMI Toshio Hokkaido University, Dept.of Maths., Assoc.Prof., 大学院・理学研究科, 助教授 (70229657)
SEKINE Jyun Osaka University, Dept of F.Enginerr., Assoc.Prof., 大学院・基礎工学研究科, 助教授 (50314399)
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| Research Abstract |
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.
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