2003 Fiscal Year Final Research Report Summary
BELLMAN EQUATIONS OF RISK-SENRSITIVE STOCHASTIC AND THEIR APPLICATIONS
| Project/Area Number |
13440033
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Osaka University |
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Principal Investigator |
NAGAI Hideo OSAKA UNIVERSITY, GRADUATE SCHOOL OF ENGINEERING SCIENCES, PROFESSOR, 基礎工学研究科, 教授 (70110848)
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| Co-Investigator(Kenkyū-buntansha) |
KIOKE Shigeaki SAITAMA UNIVERSITY, FACULTY OF EDUCATIONY, PROFESSOR, 理学部, 教授 (90205295)
SEKINE Jun OSAKA UNIVERSITY, GRADUATE SCHOOL OF ENGINEERING SCIENCES, PROFESSOR, 大学院・基礎工学研究科, 助教授 (50314399)
AIDA Shigeki OSAKA UNIVERSITY, GRADUATE SCHOOL OF ENGINEERING SCIENCES, PROFESSOR, 大学院・基礎工学研究科, 教授 (90222455)
FUNAKI Tadahisa TOKYO UNIVERSITY, GRASUATE SCHOOL OF MATHEMATICAL SCIENCE, PROFESSOR, 数理科学研究科, 教授 (60112174)
ISHII Hitoshi WASEDA UNIVERSITY, FACULTY OF EDUCATION, PROFESSOR, 教育学部, 教授 (70102887)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Bellman equations / Portfolio optimization / Maximum principle / Log Sobolev inequalities / Exponential hedge / Viscosity solutions / Critical surface models / semi-classical limits |
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| Research Abstract |
1. We considered risk-sensitive portfolio optimization problems on infinite time horizon for linear Gaussian models and general factormodels. Proving existence of solutions of ergodic type Bellman equations we got the results constructing explicitly the optimal strategies from the solutions. As for linear Gaussian models we got the same results in the case of partial information as well by only using the informations of security prices.
2. In the case of partial information, using the information of only security prices, we obtained maximum principle as necessary conditions for optimality for the problems on a finite time horizon
3. In the above case we showed that optimal strategies could be expressed explicitly by using the solution of Bellman equation with degenerate coefficients for conditionally Gaussian models
4. We showed semi-classical behavior of the minimum eigenvalues of Schrodinger operators on Wiener space can be captured in a similar way to the case of finite dimensions. By
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using similar idea we proved rough lower estimates holds for the minimum eigenvalues of the operators on path spaces (not pinned) on Riemannian manifolds. We also proved, by considering semi-classical limits on the pinned pathe space on Lie groups, that it implies that harmonic forms vanishes
5. We studied estimates of log derivatives of the heat kernels on Riemannian manifolds in which curvatures rapidly decrease enough and proved log Sobolev inequalities on path spaces. We also studied relationships between Brownian rough path and weak type poincare inequalities.
6. We studied optimization problems concerning exponential hedging in mathematical finance. In particular we calculated asymptotic expansion of the backward stochastic differential equations with respect to small parameter and obtained asymptotics of the optimal controls
7. We constructed optimal portfolio by getting higher order differentiability of the solutions of nonlinear partial differential equations arising from mathematical finance
8. We got interested in solving optimization problem by the methods of convex duality in mathematical finance and extended known. results in applying the methods to the case of partial information, or super hedging under constraints with respect to delta
9. We got the results on exsistence and uniqueness of viscosity solutions by deriving Euler equations as singular limits of minimum elements of minimization problems of functionals topologically equivalent. We got the Holder estimates of Lp viscosity solutions of fully nonlinear elliptic partial differential equation with super-linear growth with respect to first order derivatives.
10. We discussed hydrodynamic limits of critical surface models on walls and derived variational inequalities of evolution type. We also derived Alt-Caffarelli variational problems by proving large deviation principles for equilibrium systems of the critical surfaces with pinning. Less
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