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2000 Fiscal Year Final Research Report Summary

Risk-sensitive stochastic control and its singular limit

Research Project

Project/Area Number 10440030
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  Graduate School of Engineering Science, OSAKA UNIVERSITY Professor, 基礎工学研究科, 教授 (70110848)

Co-Investigator(Kenkyū-buntansha) TAKEDA Masayoshi  Tohoku Univesity, Graduate School of Science, Professor, 理学研究科, 教授 (30179650)
SEKINE Jun  Graduate School of Engineering Science, OSAKA UNIVERSITY Lecturer, 基礎工学研究科, 講師 (50314399)
AIDA Shigeki  Graduate School of Engineering Science, OSAKA UNIVERSITY Associate Professor, 基礎工学研究科, 助教授 (90222455)
FUJIWARA Tsukasa  Hyogo Education University, Associate Professor, 助教授 (30199385)
TAKANOBU Satoshi  Kanazawa University, Department of Mathematics, Associate Professor, 理学部, 助教授 (40197124)
Project Period (FY) 1998 – 2000
Keywordsrisk-sensitive stochastic control / ergodic Bellman equation / portfolio optimization / Schrodinger operator / large deviation principle / log Sobolev inequality / spectral gap / additive functional
Research Abstract

(a) It is important to know the conditions for no breakdown in risk-sensitive stochastic control problems since the value function has not always a finite value. In this research we have obtained the condition as that of the size of risk-sensitive parameter in the case of a finite time horizon and also shown the solvability of the corresponding Bellman equation under the condition. It is applicable for the case that the risk-sensitive parameter is large and has great meaning in application. In the case of infinite time horizon we have shown existence and uniqueness of the corresponding ergodic type Bellman equation under a similar condition by noticing the relationships between the problems and the eigenvalue problems for Schrodinger operator. Besides, we have derived a first order partial differential equation relating to game theoretical approach to nonlinear H_∞ control as its singular limit.
(b) We have considered portfolio optimization problem for a factor model as application to m … More athematical finance of risk-sensitive stochastic control and got the results giving the explicit representation to optimal portfolio for the problem in the case of partial information. In the case of infinite time horizon we obtained the condition under which the solution of corresponding ergodic type Bellman equation defines the optimal portfolio and constructed it by the solution. We have also found that the solution does not always give the optimal portfolio without any condition.
(c) We have shown existence of the spectral gap of Schrodinger opeartor by using log Sobolev inequality and gave the estimate.
(d) We have shown that the large deviation principle holds for additive functional of Brownian motion corresponding to measures in Kato class. As its application we obtained necessary and sufficient condition under which additive functionals converges exponentially fast.
(e) We have shown Trotter product formula with respect to L_p and trace norm and their error estimates for the Schrodinger operator with a potential bounded below.
(f) We have shown that the sequence of symmetric statistics defined by Weyl transformation on infinite dimensional torus converges to the limit represented by multiple Wiener integral under the probability measures. Less

  • Research Products

    (60 results)

All Other

All Publications (60 results)

  • [Publications] H.Nagai: "Risk-sensitive dynamic asset management with partial information"Stochastics in finite and infinite dimensions,Eds.Hida et al.,Birkhauser,Boston. 321-340 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Nagai: "Risk-sensitive potfolio optimization with partial information"Proceedings of the 39-th CDC Conference,Sydney. 1206-1211 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Bensoussan: "Conditions for no breakdown and Bellman equations of Risk-sensitive control "Applied Mathematics and its Optimization. 42. 91-101 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [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
  • [Publications] H.Kaise: "Ergodic type Bellman equations of risk-sensitive control"Proceedings of the 31^<st> ISCIE International Symposium on Stochastic Systems Theory and Its Applications. 89-94 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Takeda: "L^P independence of the spectral radius of symmetric Markov semigroups "Stochastic Processes,Physics and Geometry : New Interplays.II Eds.Fritz Gesztesy, et al.. (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Aida: "Logarithmic deribatives of heat kernels and logarithmic Sobolev inequalities with unbounded diffusion coefficients on loop space"Journal of Functional Analysis. 174. 430-477 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Aida: "Equivalence of heat kernel measure and pinned Wiener measure on loop groups"C.R.Acad.Sci.Paris. 331. 709-712 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Aida: "Precise Gaussian lower bounds on heat kernels"Stochastics in Finite and Infinite Dimensions,Eds.Hida et al.,Birkhauser,Boston. 1-28 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Aida: "On the irreducibility of the Dirichlet forms in infinite dimensional spaces "Osaka J.Math.. 37. 953-966 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ichinose: "The norm estimate of the difference between the Kac operator and Schrodinger semigroup II : The general case including the relativistic case"Electronic Journal of Probability Theory. 5. (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Sugita: "A limit theorem for Weyl transformation in infinite-dimensional torus and central limit theorem for correlated multiple Wiener integrals"Journal of Mathematical Science The University of Tokyo. 7. 99-146 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Sugita: "Random Weyl sampling for robust numerical integration of complicated functions"Monte Carlo Methods and Appl.. 6. 27-48 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Nagai: "Singular Limits of Bellman Equations of Ergodic Type Related to Risk-Sensitive Control"Stochastic Analysis,Control,Optimization and Applications,Eds.W.M.MeEneaney et al.,Birkhauser,Boston. 95-114 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Kaise: "Ergodic type Bellman equations of risk-sensitive control with large parameter"Asymptotic Analysis. 20. 279-299 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Shirakawa: "Evaluation of the asian option by the dual martingale measure"Asia-Pacific Financial Markets. 6. 183-194 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Shirakawa: "Evaluation of yieldd spread for credit risk"Mathematical Economics. 6. 83-97 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Takeda: "Exponential decay of life time and a Theorem of Kac on total occupation times"Potential Analysis. 11. 235-247 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Fukushima: "Large deviations and LIL's for Brownian motions on nested fractals"Osaka J.Math.. 36. 497-537 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Kaise : "Bellman-Isaacs equations of ergodic type related to risk-sensitive control "Asymptotic Analysis. 16. 347-362 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Beusoussan : "Some results on risk-sensitive control with full observation"Appl.Math.Optim.. 37. 1-41 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ichinose: "The norm estimate of the difference between the Kac operator and the Schrodinger semigroup"Nagoya Math.J.. 149. 53-81 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Takanobu: "On the trace norm estimate of the Trotter product formula for Schrodinger operator"Osaka J.Math.. 35. 659-682 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Sugita: "Limit theorem for symmetric statistics with respect to Weyl transformation : Disappearance of dependency "J.Math.Kyoto Univ.. 38. 653-671 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Takeda: "Asymptotic properties of generalized Feynman-Kac functionals"Potential Analysis. 9. 261-291 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 長井英生: "確率微分方程式"共立出版. 227頁 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Aida: "Short time asymptotics of certain infinite dimensional diffusion processes, (with Hiroshi Kawabi)"the Proceedings of 7th Workshop on stochastic analysis and related topics, Progress in Probability, Birkhauser.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "On the short time asymptotics of transition probability of diffusion on path groups, (with T-S.Zhang)"Potential Analysis.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "An estimate of the gap of spectrum of Schrodinger operators which generate hyperbourded semigroups"J.Funct.Anal.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "Logarithmic derivatives of heat kernels and logarithmic Sobolev inequalities with unbounded diffusion coefficients on loop spaces"J.Funct.Anal.. 174. 430-477 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "Equivalence of heat kernal measure and pinned Wiener measure on loop groups, (with Bruce Driver)"C.R.Acad.Sci.Paris. t.331, Serie I. 709-712 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "An estimate of the gap of spectrum of Schrodinger operators which generate hyperbounded semigroups"J.Funct.Anal.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "Precise Gaussian lower bounds on heat kernels"in "Stochastics in Finite and Infinite Dimensions, In bonor of Gopinath Kallianpur, " Birkhauser.. 1-28. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Aida: "On the irreducibility of the Dirichlet forms in infinite dimensional spaces"Osaka J.Math.. 37. 953-966 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Bensoussan and H.Nagai: "Conditions for no breakdown and Bellman equatuins of Risk-sensitive control"Appl.Math.Optim. 42. 91-101 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Nagai: "Risk-sensitive dynamic asset management with partial information."Stochastics in finite and infinite dimensions, Eds.Hida et al.Birkhauser, Boston. 321-340 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Nagai: "Risk-sensitive portfolio optimization with partial information"Proceedings of the 39-th CDC conference, Sydney. 1206-1211 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kuroda and H.Nagai: "Ergodic type Bellman equation of risk sensitive control and portfolio optimization on infinite time horizon"Optimal Control an Partial Differental 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
  • [Publications] H.Nagai: "Risk-sensitive stochastic control and mathematical finance, (in Japanese)"systems/control/information. vol.44, No.8. 447-454 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kaise and H.Nagai: "Ergodic Type Bellman equations of risk-sensitive control"Proceedings of The 31st ISCIE International Symposium on Stochastic System Theory and Its Applications. 89-94 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Nagai: "Singular Limits of Bellman Equations of Ergodic Type Related to Risk-Sensitive Control, Stochastic Analysis, Control"Optimization and Applications, a volume in honor of W.H.Fleming, Eds.W.M.McEneaney et al., Birkhauser. 95-114 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kaise and H.Nagai: "Ergodic Type Bellman equations of risk-sensitive control with large parameters and their singular limits"Asymptotic Analysis. 20. 279-299 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Bensoussan, J.Frehse and H.Nagai: "Some Results on Risk-sensitive with full observation"Appl.Math. and its Optimization. vol.37. 1-41 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Kaise and H.Nagai: "Bellman-Isaacs equations of ergodic type related to risk sensitive control"Asymptotic Analysis. 16. 347-362 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Sekine: "Information Geometry for Symmetric Diffusions"Potential Analysis. 1-30 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Shirakawa: "Evaluation of the asian option by the dual martingale measure"Asia-Pacific Financial Markets. 6. 183-194 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Shirakawa: "Evaluation of yield spread for credit risk"Advances in Mathematical Economics. 1. 83-97 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Ichinose and Satoshi Takanobu: "The norm estimate of the difference between the Kac operator and Schrodinger semigroup II : The general case including the relativistic case"Electronic J.Probab.. 5. 5 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi Sugita and Satoshi Takanobu: "A limit theorem for Weyl transformation in infinite-dimensional torus and central limit theorem for correlated multiple Wiener integrals"J.Math.Sci.Univ.Tokyo. 7. 99-146 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi Sugita and Satoshi Takanobu: "Random Weyl sampling for robust numerical integration of complicated fimctions"Monte Carlo Methods and Appl.. 6. 27-48 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Ichinose and Satoshi Takanobu: "The norm estimate of the difference between the Kac operator and the Schrodinger semigroup : A unified approach to the nonrelativistic and relativistic cases"Nagoya Math.J.. 149. 53-81 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Satoshi Takanobu: "On the trace norm estimate of the Trotter product formula for Schrodinger operators"Osaka J.Math.. 35. 659-682 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi Sugita and Satoshi Takanobu: "Limit theorem for symmetric statistics with respect to Weyl transformation : Disappearance of dependency"J.Math.Kyoto Univ.. 38. 653-671 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Takeda: "L^p-independence of the spectral radius of symmetric Markov semigroups, Stochastic Processes, Physics and Geometry : New Interplays. II"A Volume in Honor of Sergio Albeverio, Edited by Fritz Gesztesy, et.al. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Takeda: "Some variational formulas on additive functionals of symmetric Markov chains"Proc.Amer.Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Takeda: "Exponential decay of lifetime and a Theorem of Kac on total occupation times"Potential Analysis. 11. 235-247 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Fukushima, T.Shima and M.Takeda: "Large deviations and LIL's for Brownian motions on nested fractals"Osaka J.Math.. 36. 497-537 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Takeda: "Topics on Dirichlet forms and symmetric Markov processes"Sugaku Expositions. 12. 201-222 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Takeda: "Asymptotic properties of generalized Feynman-Kac functionals"Potential Analysis. 9. 261-291 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Nagai: "Stochastic differential equations, (in Japanese)"Kyoritsu shuppan. 227 (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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