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Study on Applications of Backward Stochastic Differential Equations

Research Project

Project/Area Number 19K03636
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionOsaka University

Principal Investigator

Sekine Jun  大阪大学, 基礎工学研究科, 教授 (50314399)

Project Period (FY) 2019-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2021: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords後退確率微分方程式 / 後退確率差分方程式 / XVA / 動的リスク尺度 / 非線形条件付き期待値 / 非線形マルコフ連鎖 / 非線形最適停止問題 / 弱時間整合性 / スパースグリッド / ランダムウォーク近似 / 部分積分公式 / 結晶格子 / マルコフ連鎖近似 / 歪曲確率 / 時間整合性 / Deep BSDE
Outline of Research at the Start

後退確率微分方程式 (Backward Stochastic Differential Equation; BSDE と略記) に関して以下を目指す.
(1) 金融実務におけるXVA(X-Valuation Adjustment)の計算に関してBSDEを用いた理論的アプローチを採用し, 理論的整備と数値解法の改良の研究を推進する.
(2) Markov型BSDEの数値解法に深層学習の技術を組み合わせたDeep BSDE手法の研究を発展させる.
(3) 金融・ファイナンス以外の分野でのBSDEの応用研究を発展させる.

Outline of Final Research Achievements

1) Using BSDEs(Backward Stochastic Differential Equations), XVA(X-Valuation Adjustment) for OTC derivative securities are studied from a mathematical finance point of view, and have obtained the following results: (i) mathematical model for financial markets are generalized. (ii) A sharper sufficient condition to ensure the No-arbitrage opportunities is obtained. (iii) Interesting examples for the existence of arbitrage opportunities are provided. (iv) An approximated computational method using an asymtotic expansion is provided, and practioners' method for computing XVA is well-explained from a theoretical point of view.
2) Backward stochastic difference equations driven by random walks on crystal lattices is studied to numerically approximate the solution of BSDEs. Convergence speed is computed and the computational error is quantitatively analyzed.

Academic Significance and Societal Importance of the Research Achievements

1)金融デリバティブの定量評価においてインパクトの大きいXVA計算について、後退確率微分方程式理論を用いて、理論整備や一般的数理的モデルの提案が行われたことに意義がある。また、最も基本的な性質:「無裁定条件」が保証されるためのパラメータ条件や逆に裁定機会が存在しうる条件の導出が行われたことは重要である。さらに、金融実務で行われている簡便法の限界や改良の提案等も行われ、数学理論と金融実務の融合が図られた。
2)より簡便な離散時間離散状態を持つ後退確率差分方程式モデルを提案し、このモデルに基づいた簡易な数値計算手法の構築と、連続時間連続状態モデルとの数値計算誤差の定量的評価が行われた。

Report

(4 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (12 results)

All 2022 2021 2020 2019

All Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 7 results,  Open Access: 1 results) Presentation (5 results) (of which Int'l Joint Research: 3 results,  Invited: 3 results)

  • [Journal Article] Notes on Backward Stochastic Differential Equations for Computing XVA2022

    • Author(s)
      Jun Sekine and Akihiro Tanaka
    • Journal Title

      Mathematics for Industry (Cheng, J., Dinghua, X., Saeki, O., Shirai, T. (eds) Proceedings of the Forum "Math-for-Industry" 2018)

      Volume: 35 Pages: 15-50

    • DOI

      10.1007/978-981-16-5576-0_2

    • ISBN
      9789811655753, 9789811655760
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On optimal thresholds for pairs trading in a one-dimensional diffusion model2021

    • Author(s)
      Masaaki Fukasawa, Hitomi Maeda, and Jun Sekine
    • Journal Title

      The ANZIAM Journal, Special Issue for Financial Mathematics, Probability and Statistics

      Volume: 63 Pages: 104-122

    • DOI

      10.21914/anziamj.v63.15437

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Backward Stochastic Differential Equations and Their Applications (IV)2019

    • Author(s)
      関根 順
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics

      Volume: 29 Issue: 4 Pages: 30-35

    • DOI

      10.11540/bjsiam.29.4_30

    • NAID

      130007823836

    • ISSN
      2432-1982
    • Year and Date
      2019-12-20
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Backward Stochastic Differential Equations and Their Applications (III)2019

    • Author(s)
      関根 順
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics

      Volume: 29 Issue: 3 Pages: 28-33

    • DOI

      10.11540/bjsiam.29.3_28

    • NAID

      130007775874

    • ISSN
      2432-1982
    • Year and Date
      2019-09-25
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Backward Stochastic Differential Equations and Their Applications (II)2019

    • Author(s)
      関根 順
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics

      Volume: 29 Issue: 2 Pages: 31-36

    • DOI

      10.11540/bjsiam.29.2_31

    • NAID

      130007720867

    • ISSN
      2432-1982
    • Year and Date
      2019-06-25
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Backward Stochastic Differential Equations and Their Applications (I)2019

    • Author(s)
      関根 順
    • Journal Title

      Bulletin of the Japan Society for Industrial and Applied Mathematics

      Volume: 29 Issue: 1 Pages: 35-40

    • DOI

      10.11540/bjsiam.29.1_35

    • NAID

      130007670619

    • ISSN
      2432-1982
    • Year and Date
      2019-03-26
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stochastic modelling with randomized Markov bridges2019

    • Author(s)
      Andrea Macrina and Jun Sekine
    • Journal Title

      Stochastics

      Volume: Online Publication Issue: 1 Pages: 1-33

    • DOI

      10.1080/17442508.2019.1703988

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Backward stochastic difference equation driven by multidimensional random walk on a lattice: convergence analysis via Wasserstein central limit theorem2021

    • Author(s)
      Jun Sekine
    • Organizer
      Centre for Financial Mathematics Seminar, University of Wollongong, Australia
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] BSDEs driven by multi-dimensional random walk on a lattice: convergence analysis via Wasserstein Central Limit Theorem2021

    • Author(s)
      田中章宏
    • Organizer
      第17回日本応用数理学会研究部会連合発表会
    • Related Report
      2020 Research-status Report
  • [Presentation] Remarks on Arbitrages in Bilateral Derivative Trading with Repo Markets2020

    • Author(s)
      Jun Sekine
    • Organizer
      The Second International Symposium on Partial Differential Equations & Stochastic Analysis in Mathematical Finance (Tsinghua International Mathematics Conference Center, Sanya, China)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 条件付歪曲期待値の時間整合性:非均一な分散構造を持つモデルの解析2020

    • Author(s)
      関根 順
    • Organizer
      日本応用数理学会 第16回研究部会連合発表会
    • Related Report
      2019 Research-status Report
  • [Presentation] The XVA issues and related BSDEs2019

    • Author(s)
      Jun Sekine
    • Organizer
      SIAM Conference on Control and Optimization
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2019-04-18   Modified: 2023-01-30  

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