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New higher order discretization method with Malliavin calculus

Research Project

Project/Area Number 16K13773
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionHitotsubashi University

Principal Investigator

Yamada Toshihiro  一橋大学, 大学院経済学研究科, 准教授 (50754701)

Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsマリアバン解析 / 確率微分方程式 / 弱近似 / 確率微分方程式の弱近似 / 高次離散化法
Outline of Final Research Achievements

In the project, we studied weak approximation schemes for stochastic differential equations driven by Brownian motions. As a result, we obtained a new higher order algorithm, the Malliavin Monte Carlo, which is constructed by some polynomials of Brownian motions through the Malliavin calculus. Not only for the standard weak approximation problem, a higher order discretization scheme for the differentiations of the solutions to partial differential equations is also obtained. We further provide another type of weak approximation using a Markov chain approximation. These results are published in major academic journals in the areas of applied mathematics such as numerical analysis, statistics and finance.

Academic Significance and Societal Importance of the Research Achievements

自然科学・社会科学においてランダムな微分方程式によるモデルは頻繁に現れ、さらにそれらのモデルの対象の平均的な振る舞いや確率計算は明らかにしたい現象を解析する上で本質的に重要である。モデルが単純であれば解析を行うことは簡単であるが、一般にはモデルは複雑になりうる。それは自然科学・社会科学の問題が単純でなく複雑だからである。本研究の成果はこのような問題に寄与しうる。従来のシミュレーション法は収束が遅いが、本研究の方法ではマリアバン解析を用いた補正を加えるので収束が非常に速くなり、また感覚的にも分かりやすい。物理現象の解析、機械学習、金融商品の数値計算やリスク管理など多くの分野への応用が期待される。

Report

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

    (13 results)

All 2019 2018 2017 2016

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Acknowledgement Compliant: 2 results) Presentation (7 results) (of which Int'l Joint Research: 3 results,  Invited: 4 results)

  • [Journal Article] Second order discretization of Bismut-Elworthy-Li formula: application to sensitivity analysis2019

    • Author(s)
      Toshihiro Yamada, Kenta Yamamoto
    • Journal Title

      SIAM/ASA Journal on Uncertainty Quantification

      Volume: 7 Issue: 1 Pages: 143-173

    • DOI

      10.1137/17m1142399

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An arbitrary high order weak approximation of SDE and Malliavin Monte Carlo: analysis of probability distribution functions2019

    • Author(s)
      Toshihiro Yamada
    • Journal Title

      SIAM Journal on Numerical Analysis

      Volume: 57 Issue: 2 Pages: 563-591

    • DOI

      10.1137/17m114412x

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A second order weak approximation of SDEs using a Markov chain without Levy area simulation2018

    • Author(s)
      Toshihiro Yamada, Kenta Yamamoto
    • Journal Title

      Monte Carlo Methods and Applications

      Volume: 24 Issue: 4 Pages: 289-308

    • DOI

      10.1515/mcma-2018-2024

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A second order discretization with Malliavin weight and Quasi-Monte Carlo method for option pricing2018

    • Author(s)
      Toshihiro Yamada, Kenta Yamamoto
    • Journal Title

      Quantitative Finance

      Volume: Latest Articles Issue: 11 Pages: 1825-1837

    • DOI

      10.1080/14697688.2018.1430371

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] A higher order weak approximation scheme of multidimensional stochastic differential equations using Malliavin weights2017

    • Author(s)
      Toshihiro Yamada
    • Journal Title

      Journal of Computational and Applied Mathematics

      Volume: 321 Pages: 427-447

    • DOI

      10.1016/j.cam.2017.03.001

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A weak approximation with Malliavin weights for local stochastic volatility model2017

    • Author(s)
      Toshihiro Yamada
    • Journal Title

      International Journal of Financial Engineering

      Volume: 04 Issue: 01 Pages: 17500021-17

    • DOI

      10.1142/s2424786317500025

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Second order discretization of Bismut-Elworthy-Li formula and applications2019

    • Author(s)
      Toshihiro Yamada
    • Organizer
      Stochastic Processes and Related Topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Second order discretization of Bismut-Elworthy-Li formula: application to sensitivity analysis2018

    • Author(s)
      Toshihiro Yamada
    • Organizer
      一橋大学 経済統計ワークショップ
    • Related Report
      2018 Annual Research Report
  • [Presentation] Higher order discretizaion methods using Malliavin Monte Carlo and Brownian Markov chain without Levy area simulation2018

    • Author(s)
      Toshihiro Yamada
    • Organizer
      WORKSHOP ON "MATHEMATICAL FINANCE AND RELATED ISSUES"
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Weak Milstein scheme without commutativity condition and its sharp asymptotic error bound2017

    • Author(s)
      山田俊皓
    • Organizer
      一橋大学経済統計ワークショップ
    • Related Report
      2017 Research-status Report
  • [Presentation] A second order discretization method for the Delta2017

    • Author(s)
      Toshihiro Yamada
    • Organizer
      Osaka-UCL Workshop on Stochastics, Numerics and Risk
    • Place of Presentation
      大阪大学(大阪府豊中市)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On higher order weak approximation with Malliavin weights2016

    • Author(s)
      Toshihiro Yamada
    • Organizer
      一橋大学 ICS FS ファカルティセミナー
    • Place of Presentation
      一橋大学 国際企業戦略科(東京都千代田区)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A general formula for weak approximation with multidimensional Malliavin weights: application to option pricing2016

    • Author(s)
      Toshihiro Yamada
    • Organizer
      一橋大学 経済統計ワークショップ
    • Place of Presentation
      一橋大学 経済学研究科(東京都国立市)
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2020-03-30  

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