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2002 Fiscal Year Final Research Report Summary

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)
Project Period (FY) 2001 – 2002
KeywordsHormander condition / Malliavin calculus / filtering equation / jump type process / hypoellipticity / Hilbert space / stochastic flow / stochastic differential equation
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.

  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] T.Komatsu: "On the Malliavin calculus for Markov processes with jumps,"Probability Theory and Mathematical Statistics (Proceedings). Kyiv,2002. 59-72 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Komatsu: "On the Malliavin calculus for stochastic flows with interaction on Hilbert spaces,"Proceedings of the 8th Vilnius Conference on Prob. Theory and Math. Stat. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [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(和文)」より
  • [Publications] A.Takeuchi: "Mailiavin calculus for SDE with jumps and the partially hypoelliptic problem,"Osaka J. Math.. 39. 523-559 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Kamae: "Stochastic analysis based on deterministic Brownian motion"Israel J. Math.. 125. 317-346

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Yoshida: "On the pinched circle model and the absence of wandering domains for a topological polynomial,"Indagationes Mathematicae. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Komatsu: "On the Malliavin calculus for stochastic flows with interaction on Hilbert spaces"Proceedings of the 8th Vilnius Conference on Prob. Theory and Math. Statistics, 2002 Vilnius. to appear. (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Komatsu: "On the Malliavin calculus for Markov processes with jumps"Probability Theory and Mathematical Statistics (Proceedings of Ukrainian Mathematical Congress - 2001), Kyiv. 59-72 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Komatsu, A. Takeuchi: "Generalized Hormander theorem for non-local operators"Proceeding of the first Sino-German Conference on Stochastic Analysis, 2002 Beijing. to appear. (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [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
  • [Publications] T. Komatsu, A. Takeuchi: "Simplified probabilistic approach to the Hormander theorem"Osaka J. Math.. 38. 681-691 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Takeuchi: "On the partially hypoelliptic problem via Malliavin calculus"Probability Theory and Mathematical Statistics (Proceedings of Ukrainian Mathematical Congress - 2001), Kyiv. 160-169 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Takeuchi: "The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem"Osaka J. Math.. 39. 523-559 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Yoshida: "On the pinched circle model and the absence of wandering domains for a topological polynomial"Indagationes Mathematicae. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Yoshida: "Gap invariance of a symmetric invariant lamination"Tokyo J. Math.. 24. 1-12 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Kamae: "Stochastic analysis based on deterministic Brownian motion"Israel J. Math.. 125. 317-346 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Kamae, L. Zamboni: "Sequence entropy and the maximal pattern complexity of infinite words"Ergodic Theory and Dynamical Systems. 22. 1191-1199 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Kamae, L. Zamboni: "Maximal pattern complexity for discrete systems"Ergodic Theory and Dynamical Systems. 22. 1201-1214 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Kamae, et al.: "Automata, algebraicity and distribution of sequences of powers"Annales de Institut Fourier. 51. 687-705 (2001)

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      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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