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Malliavin calculus for stochastic differential equations with jumps

Research Project

Project/Area Number 13640132
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, 大学院・理学研究科, 教授 (80047365)

Co-Investigator(Kenkyū-buntansha) DATEYAMA Masahito  Osaka City Univ., Science, Lecturer, 大学院・理学研究科, 講師 (10163718)
FUJIHARA Tsukasa  Hyogo Univ. Edu., Education, Assoc. Prof., 学校教育学部, 助教授 (30199385)
KAMAE Tetsuro  Osaka City Univ., Science, Professor, 大学院・理学研究科, 教授 (80047258)
TAKEUCHI Atsushi  Osaka City Univ., Science, Res. Assoc., 大学院・理学研究科, 助手 (30336755)
YOSHIDA Masamichi  Osaka City Univ., Science, Lecturer, 大学院・理学研究科, 講師 (60264793)
根来 彬  静岡大学, 工学部, 教授 (80021947)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsHormander condition / Malliavin calculus / filtering equation / jump type process / hypoellipticity / Hilbert space / stochastic flow / stochastic differential equation / 飛躍型確立過程 / フィルタリング方程式 / セミマルチンゲール / ヘビィ過程 / 確率密度関数
Research Abstract

The first result is on the regularity of solution to the filtering equation for a certain system of jump type processes. Applying a new key lemma in the Malliavin calculus, the existence of smooth density of the solution was proved under a generalized Hormander condition. The second result is the following :
Let a_0(x,x^^-) be an R^N - valued smooth function on R^N × R^N , v(dθ) be a finite measure on a discrete space θ, β_t = (β^θ_t) be a Wiener process, and let μ(dv) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u ∈ R^d, 【numerical formula】
Assume that x^u(0) is smooth in u and the process x(t) = (x^u(t)) takes its values in the Hilbert space H = (L^2(R^d,B(R^d),μ))^N. Let π: H → R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H : 【numerical formula】
The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H with the help of a new key lemma. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report

Research Products

(31 results)

All Other

All Publications

  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2002 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report
  • [Publications]

    • Related Report
      2001 Annual Research Report

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

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