• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

BELLMAN EQUATIONS OF RISK-SENRSITIVE STOCHASTIC AND THEIR APPLICATIONS

Research Project

Project/Area Number 13440033
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionOsaka University

Principal Investigator

NAGAI Hideo  OSAKA UNIVERSITY, GRADUATE SCHOOL OF ENGINEERING SCIENCES, PROFESSOR, 基礎工学研究科, 教授 (70110848)

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)
藤田 安啓  富山大学, 理学部, 助教授 (10209067)
竹田 雅好  東北大学, 理学研究科, 教授 (30179650)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥11,900,000 (Direct Cost: ¥11,900,000)
Fiscal Year 2003: ¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 2002: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2001: ¥4,500,000 (Direct Cost: ¥4,500,000)
KeywordsBellman equations / Portfolio optimization / Maximum principle / Log Sobolev inequalities / Exponential hedge / Viscosity solutions / Critical surface models / semi-classical limits / ポートフェリオ最適化 / ハミルトン・ヤコビ方程式 / 緩和現象 / コルモゴロフ方程式 / 最適戦略 / ウィーナー空間 / 非完備市場 / ファインマン・カッツ汎関数 / リスク鋭感的確率制御 / エルゴード型ベルマン方程式 / シュレーディンガー作用素 / 大偏差原理 / スペクトルギャップ / 加法的汎関数
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 … More 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

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (64 results)

All Other

All Publications (64 results)

  • [Publications] Hideo NAGAI: "Optimal strategies for risk-sensitive portfolio optimization problems for general factor models"SIAM Journal on Control and Optimization. 41. 1779-1800 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideo NAGAI: "Risk-sensitive portfolio optimization with full and partial information"Advanced Studies of Pure Mathematics. 41. 257-278 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Shigeki Aida: "On a certain semiclassical problem on Wiener spaces"Publications of Research Institute of Mathematical Science. 39. 365-392 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Shigeki Aida: "Semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space"Journal of Functional Analysis. 203. 401-424 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Jun Sekine: "An approximation for exponential hedging"Advanced Studies in Pure Mathematics. 41. 279-299 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Koike: "Variational inequalities for leavable bounded-velocity control"Applied Mathematics and Optimizations. 48. 1-20 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kuroda: "Ergodic type Bellman equation of risk sensitive control and portfolio optimization on infinite time horizon"Optimal Control and Partial Differential Equations, Eds.Menaldi et al. IOS press, Amsterdam. 530-538 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Nagai: "Risk-sensitive optimal investment problems with partial information on infinite time horizon"Recent developments in Mathematicl finance, Ed.J.Yong, World Scientific. 85-98 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kuroda: "Risk-sensitive portfolio optimization on infinite time horizon"Stochastics and Stochastics Reports. 73. 309-332 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Nagai: "Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon"Annals of Applied Probability. 12. 1-23 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Aida: "Short time asymptotics of a certain infinite dimensional diffusion process"Stochastic analysis and related topics VII The Silivri Workshop, Progress in Probability Birkhauser. 77-124 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Aida: "An estimate of the gap of spectrum of Schrodinger operators which generate hyperbounded semigroups"J.Functional Analysis. 185. 474-526 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 仁科 一彦: "金融工学"大阪大学出版会. 88 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 舟木 直久: "ミクロからマクロへ1 界面モデルの数理"シュプリンガーフェアラーク東京. 300 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideo NAGAI: "Optimal strategies for risk-sensitive portfolio optimization problems for general factor models"SIAM Journal on Control and Optimization. 41. 1779-1800 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideo NAGAI: "Risk-sensitive portfolio optimization with full and partial information"Advanced Studies of Pure Mathematics. 41. 257-278 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Shigeki Aida: "On a certain semiclassical problem on Wiener spaces"Publications of Research Institute of Mathematical Science. 39. 365-392 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Shigeki Aida: "Semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space"Journal of Functional Analysis. 203. 401-424 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Jun Sekine: "An approximation for exponential Hedging"Advanced Studies in Pure Mathematics. 41. 279-299 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Koike: "Variational inequalities for leavable bounded-velocity control"Applied Mathematics and Optimizations. 48. 1-20 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kuroda: "Ergodic type Befman equation of risk sensitive control and portfolio optimization on infinite time horizon"Optimal Control and Partial Differential Equations (Eds. Menaldi et al.), (IOS press, Amsterdam). 530-538 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Nagai: "Risk-sensitive optimal investment problems with partial information on infinite time horizon"Recent developments in Mathematid finance. (Ed. J.Yong) (World Scientific). 85-98 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K Kuroda: "Risk-sensitive portfolio optimization on infinite time horizon"Stochastics and Stochastics Reports. 73. 309-332 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Nagai: "Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon"Annals of Applied Probability. 12. 1-23 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Aida: "Short time asymptotics of a certain infinite dimensional diffusion process"Stochastic analysis and related topics VII The Silivri workshop", Progress in Probability, Birkhauser. 77-124 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Aida: "Schrodinger operators which generate hyperbounded semigroups"J. of Functional Analysis. 185. 474-526 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazuhiko Nishina: "Financial Engineering"Osaka University press. 88 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tadahisa Funaki: "Micro to Macro 1 Mathematics of critical surface models"Springer-Verlag, Tokyo. 300 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hideo NAGAI: "Optimal strategies for risk-sensitive portfolio optimization problems for general factor models"SIAM Journal on Control and Optimization. 41. 1779-1800 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hideo NAGAI: "Risk-sensitive portfolio optimization with full and partial information"Advanced Studies of Pure Mathematics. 41. 257-278 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Aida: "On a certain semiclassical problem on Wiener spaces"Publications of Research Institute of Mathematical Science. 39. 365-392 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeki Aida: "Semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space"Journal of Functional Analysis. 203. 401-424 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Jun Sekine: "An approximation for exponential hedging"Advanced Studies in Pure Mathematics. 41. 279-299 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Koike: "Variational inequalities for leavable bounded-velocity control"Applied Mathematics and Optimizations. 48. 1-20 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Koike: "Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear, uniformly elliptic equations with measurable and quadratic terms"Nonlinear Differential Equations and Applications. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ishii: "Relaxation of Hamilton-Jacobi equations"Archiv fur Rational Mechanics and Analysis. 169. 265-304 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ishii: "Relaxation in an $L^\infty$-optimization problem"Proceedings of Royal Society of Edingburgh. 133. 599-615 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ishii: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Communication on Partial differential Equations. 28. 409-438 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Fujita: "A comparison theorem for Bellman equations of ergodic control"Differential and Integral Equations. 16. 641-651 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Fujita: "A linear PDE approach to Bellman equation of ergodic control with periodic structure"Applied Mathematics and Optimization. 47. 143-149 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kuroda: "Risk-sensitive portfolio optimization on infinite time horizon"Stochastics and Stochastics Reports. 73. 309-331 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Nagai: "Optimal strategies for risk-sensitive portfolio optimization problems for general factor models"SIAM Journal on Control and Optimization. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Aida: "On the small time asymptotics of diffusion processes on path groups"Potential analysis. 16. 67-78 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Aida: "Witten complex on pinned path group and its expected semiclassical behaviour"the special issue"Probability and Geometry" of "Infinite dimensional analysis, Quantum probability and Related topics. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Sekine: "On superhedging under delta constraints"Applied Mathematical Finance. 9. 103-121 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Takeda: "Conditional gaugeability and subcriticality of generalized Schr\"odinger operators"Journal of Functional Analysis. 191. 343-376 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Takeda: "Large deviation principle for additive functionals of Brownian motion corresponding to Kato measures"Potential Analysis. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] "Hydrodynamic limit for $\nabla \phi$ interface model on a wall"Probability Theory and Related Fields. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii: "A two-dimensional random crystalline algorithm for Gauss curvature flow"Advance in Applied Probability. 34. 491-504 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii: "A class of stochastic control problems with state constraint"Indiana Univ. Mathematical Journal. 51. 1169-1198 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 内山耕平: "ミクロからマクロへ2,格子気体の流体力学極限"シュプリンガー・フェアラーク東京. 303 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kuroda: "Ergodic type Bellman equation of risk sensitive control and portfolio optimization on infinite time horizon"Optimal Control and Partial Differential Equations, Eds. M\'enaldi et al., IOS press, Amsterdam. 530-538 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Nagai: "Risk-sensitive optimal investment problems with partial information on infinite time horizon"Recent developments in Mathematicl finance, Ed. J. Yong, World Scientific. 85-98 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kuroda: "Risk-sensitive portfolio optimization on infinite time horizon"to appear in Stochastics and Stochastics Reports.

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Nagai: "Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon"Annals of Applied Probability. 12. 1-23 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Aida: "Short time asymptotics of certain infinite dimensional diffusion process""Stochastic analysis and related topics VII The Silivri workshop", Progress in Probability, Birkha\"user. 77-124 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Aida: "An estimate of the gap of spectrum of Schr\"odinger operators which generate hyperbounded semigroups"J. Functional Analysis. 185. 474-526 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Ishibashi: "On fully nonlinear PDEs devied from variational problems of $L^p$ norms"SIAM Journal on Mathematical Analysis. 33. 545-569 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Koike: "Remarks on regularity of viscosity solutions of fully nonlinear uniformly elliptic PDEs with measurable ingredients"Advances in Differential Equations. 7. 493-512 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Takeda: "Conditional Gaugeability and Subcriticality of Generalized Schr\"odinger Operators"to appear in J. Funct.Anal.

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Funaki: "Large deviations for the Ginzburg-Landau interface model"Probab. Theory Relat. Fields. 120. 535-568 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Funaki: "Fluctuations for $\nabla \phi$ interface model on a wall, Stoch. Proc. Appl."Stoch. Proc. Appl.. 94. 1-27 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] D.H.Kim: "Some variational formulae in additive functionals of symmetric Markov chains"To appear in Proc. Amer. Math. Soc..

    • Related Report
      2001 Annual Research Report
  • [Publications] 舟木 直久: "ミクロからマクロヘ1 界面モデルの数理"シュプリンガーフェアラーク東京. 300 (2002)

    • Related Report
      2001 Annual Research Report

URL: 

Published: 2001-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi