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Study on regularities of stochastic processes with jumps

Research Project

Project/Area Number 11640133
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KOMATSU Takashi  Osaka City Univ., Science, Professor, 理学部, 教授 (80047365)

Co-Investigator(Kenkyū-buntansha) DATEYAMA Masahito  Osaka City Univ., Science, Lecturer, 理学部, 講師 (10163718)
KAMAE Tetsuro  Osaka City Univ., Science, Professor, 理学部, 教授 (80047258)
NEGORO Akira  Shizuoka Univ., Technology, Professor, 工学部, 教授 (80021947)
HIRABA Seiji  Osaka City Univ., Science, Lecturer, 理学部, 講師 (30260798)
YOSHIDA Masamichi  Osaka City Univ., Science, Res. Assoc., 理学部, 助手 (60264793)
藤井 準二  大阪市立大学, 理学部, 講師 (60117968)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥1,500,000 (Direct Cost: ¥1,500,000)
KeywordsHormander theorem / Malliavin calculus / pseudo-differential operator / hypoellipticity / transition density / Markov process / jump type process / stochastic differential equation / 確率密度関数 / セミマルチンゲール
Research Abstract

We studied on the Hormander theorem via the Malliavin calculus for the parabolic pseudo-differential operator :
<<numerical formula>>
where <<numerical formula>>, and B_θ are operators defined by <<numerical formula>>. The hypoellipticity of the above operator is equivalent to the existence of smooth densities of transition probabilities of Markov processes with jumps which are solutions to stochastic integro-differential equations :
<<numerical formula>>
where β(s)=(β^κ(s)) is an m-dimensional Brownian motion and J(dsdθ) is a Poisson random measure with E[J(dsdθ)]=π(dθ)ds. A similar problem for usual parabolic differential operators was studied in the course of the Malliavin calculus.
We carried out the variation for jump type Markov processes by Girzanov transforms of Levy processes, and proved special necessary formulas of integration by parts on the cad-lag space. It must be shown the strong decay of the Laplace transform of the distribution of a specific functional associated with the Malliavin covariance. So far, similar strong decay property was proved by long complicated arguments. But we proved it by a new simple method where the key lemma is an estimate for general semimartingales. We proved the smoothness of transition densities of Markov processes with jumps under certain conditions which are essentially weaker than the Hormander type condition introduced by Leandre.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] T.Komatsu,A.Takeuchi: "On the smoothness of pdf of solutions to SDE of jump type"International J.Differential Equations and Appl. 2. 141-197 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Komatsu,A.Takeuchi: "Simplified probabilistic approach to the Hormander theorem"Osaka J.Math.. 38(to appear). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Negoro,K.Kikuchi: "On Markov process generated by pseudodifferential operator of variable order"Osaka J.Math.. 34. 319-335 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kamae: "Stochastic analysis based on deterministic Brownian motion"Israel J.Math.. (to appear). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Dateyama,T.Kamae: "On direct sum decomposition of integers and Ito's conjecture"Tokyo J.Math.. 21. 433-440 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Yoshida: "Gap invariance of a symmetrie invariant lamination"Tokyo J.Math.. 24(to appear). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Hiraba: "Jump-type Fleming-Viot Processes"Advances in Applied Probability. 32. 101-122 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Komatsu, A.Takeuchi: "On the smoothness of pdf of solutions to SDE of jump type"International J.Differential Equation and Appl.. 2. 141-197 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Komatsu, A.Takeuchi: "Simplified probabilistic approach to the Hormander theorem"Osaka J.Math.. 38 (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Negoro, K.Kikuchi: "On Markov process generated by pseudodifferential operator of variable order"Osaka J.Math.. 34. 319-335 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kamae: "Stochastic analysis based on deterministic Brownian motion"Israel J.Math.. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Dateyama, T.Kamae: "On direct sum decomposition of integers and Y.Ito's conjecture"Tokyo J.Math.. 21. 433-440 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Yoshida: "Gap invariance of a symmetric invariant lamination"Tokyo J.Math.. 24 (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Hiraba: "Jump-type Fleming-Viot processes"Advances in Applied Probability. 32. 101-122 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Komatsu,A.Takeuchi: "On the smoothness of pdf of solutions to SDE of jump type"International J. Differential Equations and Applications. 2. 141-197 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Komatsu,A.Takeuchi: "Simplified probabilistic approach to the Hormander theorem"Osaka J. Math. 38(to appear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Negoro,K.Kikuchi: "On Markov process gererated by pseudodifferential operator of variable…"Osaka J. Math.. 34. 319-335 (1997)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kamae: "Linear expansions, strictly ergodic homogeneous cocycles and fractals"Israel J. Math.. 106. 313-337 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Dateyama,T.Kamae: "On direct sum decomposition of integers and Ito's conjecture"Tokyo J. Math.. 21. 433-440 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Yoshida: "On variational principle and metrics associated with a potential of…"Commentarii Mathematici Universitatis Sancti Pauli. 47. 1-5 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] T. Komatsu, A. Takeuchi: "Simplified probabilistic approach to the Hormander Theorem"Osaka J. Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] A. Negoro, K. Kikuchi: "On Markov process generated by pseudodifferential oprator・・・"Osaka J. Math.. 34. 319-335 (1997)

    • Related Report
      1999 Annual Research Report
  • [Publications] T. Kamae: "Linear expansions, strictly ergodic homogeneous cocycles and fractals"Israel J. Math.. 106. 313-337 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Dateyama, T. Kamae: "On direct sum decomposition of integers and Ito's conjecture"Tokyo J. Math.. 21. 433-440 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. Hiraba: "Jump-type Fleming-Viot processes"Advances in Applied Probability. 32. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Yoshida: "On variational principle and metrics associated with a potential of ・・・"Commentarii Mathematici Unicersitatis Sancti Pauli. 47. 1-5 (1998)

    • Related Report
      1999 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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