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Gauge Theory of Stochastic Processes

Research Project

Project/Area Number 15K17562
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionFukuoka University (2017-2018)
Ritsumeikan University (2015-2016)

Principal Investigator

Amaba Takafumi  福岡大学, 理学部, 講師 (10737539)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords超関数に対する伊藤の公式 / Wiener汎関数の正則化 / Malliavin解析 / 双対確率流 / Loewner-Kufarev方程式 / Grunsky係数 / tau関数 / 定常Gauss過程 / Level Crossings / Signature / Witt代数 / Skorokhod積分 / smoothing effect / 離散時間確率流 / 局所時間 / 伊藤の公式
Outline of Final Research Achievements

1. It is proven that the scheme approximating an irregular function of a Levy process by discretizing the Clark-Ocone formula with respect to time has the strong-convergence rate given by half of the Sobolev index of the functional (Joint with Nien-Lin Liu, Azmi Makhlouf and Takwa Saidaoui). 2. We have introduced a class which we call controlled Loewner-Kufarev equation. We embedded the solution to the Sato-Segal-Wilson Grassmannian accoding to the procudure of Krichever, and then we described the time-evolution in the Grassmannian and obtained an explicit formula describing the associated Grunsky coefficients (Joint with Roland Friedrich). 3. We proved that integration with respect to time of the function of a solution to a stochastic differential equation (or smooth stationary Gaussian process and its derivative) raise the Sobolev index by 1 (1/2 respectively) than that of the integrand (Joint with Yoshihiro Ryu and Kratz Marie).

Academic Significance and Societal Importance of the Research Achievements

1はオプションのペイオフ関数の近似において、その関数が滑らかでないときにも、その少ない滑らかさに応じて近似の精度を具体的に保証するという意味で実務的にも有意義である。2はこれまで確率論の文脈においては統計モデルのみに用いられていたLoewner方程式の、無限可積分系における立ち位置を示唆しているように見える、という意味で分野を横断して意義がある。3や4は機械学習の中でもGauss過程回帰モデルの文脈においてlength scaleと呼ばれるハイパーパラメータを、Rice公式とMC法を用いて推定する際に、その精度を保証することに繋がるため、このモデルを実装する社会の様々な文脈において有用である。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (22 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (8 results) Journal Article (1 results) (of which Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (13 results) (of which Int'l Joint Research: 3 results,  Invited: 1 results)

  • [Int'l Joint Research] Saarland University(ドイツ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] ESSEC Business School, CREAR(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Université de Tunis El Manar(チュニジア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Saarland university(ドイツ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] ESSEC CREAR(フランス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Universitat des Saarlandes(Germany)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Universite de Tunis El Manar/Ecole Nationale d'Ingenieurs de Tunis(Tunisia)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] CREAR(France)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals2018

    • Author(s)
      Takafumi Amaba and Yoshihiro Ryu
    • Journal Title

      ALEA, Lat. Am. J. Probab. Math. Stat.

      Volume: 15 Issue: 1 Pages: 703-703

    • DOI

      10.30757/alea.v15-27

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] On the regularity of time occupation functionals for Gaussian processes2019

    • Author(s)
      Takafumi Amaba
    • Organizer
      Okayama Workshop on Stochastic Analysis 2019
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the regularity of time occupation functionals for Gaussian processes2019

    • Author(s)
      Takafumi Amaba
    • Organizer
      Mathematical Finance seminar at Ritsumeikan university
    • Related Report
      2018 Annual Research Report
  • [Presentation] Loewner-Kufarev方程式の連続率について2018

    • Author(s)
      天羽 隆史
    • Organizer
      測地線及び関連する諸問題
    • Related Report
      2017 Research-status Report
  • [Presentation] On dynamics of some equations related to stochastic objects2018

    • Author(s)
      天羽 隆史
    • Organizer
      岡山--広島 解析・確率論セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] An integration by parts on “space of loops”2017

    • Author(s)
      Takafumi Amaba
    • Organizer
      Probability Seminar
    • Place of Presentation
      Universite du Luxembourg (ルクセンブルグ)
    • Year and Date
      2017-01-19
    • Related Report
      2016 Research-status Report
  • [Presentation] Distributional Ito^'s formula and regularization of generalized Wiener functionals2017

    • Author(s)
      天羽 隆史
    • Organizer
      確率解析シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] Loewner-Kufarev方程式の連続率について2017

    • Author(s)
      天羽 隆史
    • Organizer
      福岡大学微分幾何研究集会
    • Related Report
      2017 Research-status Report
  • [Presentation] Loewner-Kufarev方程式の連続率について2017

    • Author(s)
      天羽 隆史
    • Organizer
      九州確率論セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] Loewner-Kufarev方程式の連続率について2017

    • Author(s)
      天羽 隆史
    • Organizer
      確率論シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] An integration by parts on “space of loops”2016

    • Author(s)
      Takafumi Amaba
    • Organizer
      Oberseminar zur Freien Wahrscheinlichkeitstheorie
    • Place of Presentation
      Universitat des Saarlandes (ドイツ)
    • Year and Date
      2016-12-13
    • Related Report
      2016 Research-status Report
  • [Presentation] Holder continuity of a diffusion local time2016

    • Author(s)
      天羽隆史
    • Organizer
      確率論ヤングサマーセミナー
    • Place of Presentation
      松嶋館(三重県, 伊勢市)
    • Year and Date
      2016-08-13
    • Related Report
      2016 Research-status Report
  • [Presentation] Regularization of generalized Wiener functionals by Bochner integral2015

    • Author(s)
      Takafumi Amaba
    • Organizer
      2015年度 確率解析シンポジウム
    • Place of Presentation
      大阪府豊中市 大阪大学基礎工学国際棟セミナー室
    • Year and Date
      2015-10-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Convergence implications via dual flow method2015

    • Author(s)
      Takafumi Amaba
    • Organizer
      Stochastic Analysis and Applications German-Japanese, bilateral research project, 2015
    • Place of Presentation
      宮城県仙台市 東北大学 片平キャンパス
    • Year and Date
      2015-08-31
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research

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Published: 2015-04-16   Modified: 2020-03-30  

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