Rough path theory and its application to probability theory

• Project/Area Number: 19740064
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: INAHAMA Yuzuru Tokyo Institute of Technology, 大学院・理工学研究科, 助教 (80431998)
• Project Period (FY): 2007 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥2,170,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
• Keywords: 確率解析 / 無限次元解析 / ラフパス理論 / 確率微分方程式 / 伊藤写像 / 大偏差原理 / ラプラス近似 / 漸近展開 / 確率論 / 確率テーラー展開 / ループ空間

Research Abstract

研究成果の概要 : ラフパス理論とは、伊藤清の創設した通常の確率微分方程式論を「非ランダム化」するものであり、その結果、微分方程式の入力パスを出力パスに対応させる伊藤写像は連続写像になる。本研究においては、この伊藤写像にたいしてある種のテーラー展開を試み、それを利用して確率論の有名な漸近定理(大偏差原理やラプラス近似)を示した。またこの種の定理をバナッハ空間に値をとるプロセスに適用すると、通常の確率微分方程式論では証明できていない例が示せることがわかった。現在は、この種の定理をいろいろな場合に拡張中である。

Research Products

• [Journal Article] On the Laplace-type asymptotics and the stochastic Taylor expansion for It\^o functionals of Brownian rough paths (2008)
  • Author(s): Yuzuru Inahama & Hiroshi Kawabi
  • Journal Title: {\it RIMS Kokyuroku Bessatsu} {\bf B6} Pages: 139-152
  • Peer Reviewed
• [Journal Article] On the Laplace-type asymptotics and the stochastic Taylor expansion for It\^o functionals of Brownian rough path (2008)
  • Author(s): Yuzuru Inahama
  • Journal Title: RIMS Kokyuroku Bessats B6 Pages: 139-152
  • Peer Reviewed
• [Journal Article] On asymptotics of Banach space-valued It\~{o} functionals of Brownian rough paths (2007)
  • Author(s): Yuzuru Inahama & Hiroshi Kawabi
  • Journal Title: Stochastic Analysis and Applications -The Abel symposium 2005, Stockholm, Springer Pages: 415-434
  • Peer Reviewed
• [Journal Article] Asymptotic expansions for the Laplace approximations for It\^{o} functionals ofBrownian rough paths (2007)
  • Author(s): Yuzuru Inahama & Hiroshi Kawabi
  • Journal Title: J. Funct. Anal. {\bf 243} no. 1 Pages: 270-322
  • Peer Reviewed
• [Journal Article] Asymptotic expansions for the Laplace approximations for It\^{o} functionals of Brownian rough paths (2007)
  • Author(s): Yuzuru lnahama, Hiroshi Kawabi
  • Journal Title: J. Funct. Anal. 243 Pages: 270-322
  • Peer Reviewed
• [Journal Article] On asymptotics of Banach space-valued It\-{o} functionals of Brownian rough paths (2007)
  • Author(s): Yuzuru lnahama, Hiroshi Kawabi
  • Journal Title: The Abel symposium 2005, Stockholm, Springer Pages: 415-434
  • Peer Reviewed
• [Journal Article] On the Laplace-type asymptotics and the stochastic Taylor expansion for lt\^o functionals of Brownian rough paths (2007)
  • Author(s): Yuzuru lnahama, Hiroshi Kawabi
  • Journal Title: RIMS Kokyuroku Bessatsu B6 Pages: 139-152
  • Peer Reviewed
• [Presentation] Rough path analysis for an infinite dimensional diffusion (2009)
  • Author(s): 稲浜譲
  • Organizer: 日本数学会 統計数学分科会 特別講演
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] A stochastic Taylor-like expansion in the rough path theory. (2008)
  • Author(s): Yuzuru Inahama
  • Organizer: The 1st MSJ-SI (The Mathematical Society of Japan, Seasonal Institute), Probabilistic Approach to Geometry
  • Place of Presentation: 京都大学
  • Year and Date: 2008-07-31
• [Presentation] A stochastic Taylor-like expansion in the rough path theory (2008)
  • Author(s): Yuzuru Inahama
  • Organizer: The 1st MSJ-SI (The Mathematical Society of Japan, Seasonal Ins titute), Probabilistic Approach to Geometr
  • Place of Presentation: 京都大学
  • Year and Date: 2008-07-31
• [Presentation] A stochastic Taylor-like expansion in the rough path theory. (2007)
  • Author(s): 稲浜譲
  • Organizer: 確率解析とその周辺
  • Place of Presentation: 東京工業大学
  • Year and Date: 2007-11-24
• [Presentation] A stochastic Taylor-like expansion in the rough path theory (2007)
  • Author(s): 稲浜 譲
  • Organizer: 確率論とその周辺
  • Place of Presentation: 東京工業大学
  • Year and Date: 2007-11-24
• [Presentation] A stochastic Taylor-like expansion in the rough path theory. (2007)
  • Author(s): Yuzuru Inahama
  • Organizer: Stochastic calculus on manifolds, graphs, and random structures
  • Place of Presentation: University of Bonn
  • Year and Date: 2007-10-08
• [Presentation] A stochastic Tay1or-like expansion in the rough path theory (2007)
  • Author(s): Yuzuru Inahama
  • Organizer: Stochastic Calculus on manifolds, graphs, and Random structures
  • Place of Presentation: Univ. Bonn
  • Year and Date: 2007-10-08