Construction of new algorithms for numerical weak approximation of Diffusion Processes by Kusuoka scheme and their applications to Finance problems

• Project/Area Number: 18540113
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: NINOMIYA Syoiti Tokyo Institute of Technology, 大学院・イノベーションマネジメント研究科, 教授 (70313377)
• Co-Investigator(Kenkyū-buntansha): 楠岡 成雄 東京大学, 大学院・数理科学研究科, 教授 (00114463)
• Co-Investigator(Renkei-kenkyūsha): KUSUOKA Shigeo 東京大学, 大学院・数理科学研究科, 教授 (00114463)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,630,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥630,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 確率論 / 数理工学 / アルゴリズム / 数理ファイナンス / デリバティブ / 確率微分方程式 / 確率諭

Research Abstract

(1) We have succeeded in finding and constructing new higher-order numerical approximation algorithms for diffusion processes. The algorithms are based on the theory of Kusuoka approximation. They enjoy both the numerical robustness and the universality, that is, we can apply them for almost all diffusions. (2) We applied the algorithms to some finance problems and achieved remarkable improvements in : (a) very fast calculation (at least 100 times faster than the state of the art methods) (b) robust discretization (c) universality. (3) We have developed the computer program of the algorithms.

Research Products

• [Journal Article] A certain Limit of Iterated CTE (2010)
  • Author(s): Shigeo Kusuoka
  • Journal Title: Adv. in Math. Eco. vol. 13 Pages: 99-111
• [Journal Article] A new higher-order weak approximation scheme of stochastic differential equations and the Runge-Kutta method (2009)
  • Author(s): Mariko Ninomiya, Syoiti Ninomiya
  • Journal Title: Finance and Stochastics vol.13,No.3 Pages: 415-443
• [Journal Article] A Remark on the Asymptotic Expansion of density function of Wiener Functionals (2009)
  • Author(s): S. Kusuoka and H. Osajima
  • Journal Title: J. Fuct. Analysis 255 Pages: 2545-2562
  • Peer Reviewed
• [Journal Article] Weak approximation of stochastic differential equations and application to derivative pricing (2008)
  • Author(s): Syoiti Ninomiya, Nicolas Victoir
  • Journal Title: Applied Mathematical Finance vol.15,No.2 Pages: 107-121
• [Journal Article] Weak approximation of stochastic differential equations and application to derivative pricing (2008)
  • Author(s): Syoiti Ninomiya and Nicolas Victoir
  • Journal Title: Applied Mathematical Finance Vol.15, No.2 Pages: 107-121
  • Peer Reviewed
• [Journal Article] A new higher-order weak approximation scheme of stochastic differential equations and the Runge-Kutta method
  • Author(s): Syoiti Ninomiya and Mariko Ninomiya
  • Journal Title: Finance and Stochastics (to appear)
  • URL: http://arxiv.org/abs/0709.2434
  • Peer Reviewed
• [Presentation] Approximation of Expectation of Diffusion Processes with Dirichlet boundary (2010)
  • Author(s): Shigeo Kusuoka
  • Organizer: International Workshop on Mathematical Finance : Topics on Leading-edge Numerical Procedures and Models
  • Place of Presentation: Tokyo
• [Presentation] A higher-order weak approximation method of SDEs (2009)
  • Author(s): 二宮祥一
  • Organizer: Workshop: Computational Finance
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] A higher-order weak approximation method of SDEs and the Runge-Kutta method (2009)
  • Author(s): 二宮祥一
  • Organizer: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance"
  • Place of Presentation: Wolfgang Pauli Institute, Vienna
• [Presentation] 確率微分方程式の新しい弱近似法:楠岡近似とそれを実現するアルゴリズム (2009)
  • Author(s): 二宮祥一
  • Organizer: 計算による数理科学の展開2009(研究集会)
  • Place of Presentation: 神戸大学理学部
• [Presentation] A higher-order weak approximation method of SDEs (2009)
  • Author(s): 二宮祥一
  • Organizer: Workshop : Computational Finance
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] A higher-order weak approximation method of SDEs and the Runge-Kutta method (2009)
  • Author(s): Syoiti Ninomiya
  • Organizer: Conference on small time asymptotics, perturbation theory and heat kernel methods in mathematical finance
  • Place of Presentation: Wolfgang Pauli Institute, Vienna
• [Presentation] 確率微分方程式の新しい弱近似法 : 楠岡近似とそれを実現するアルゴリズム (2009)
  • Author(s): 二宮祥一
  • Organizer: 計算による数理科学の展開2009(研究集会)於神戸大学理学部
  • Place of Presentation: 神戸
• [Presentation] 確率微分方程式の新しい弱近似法 : 楠岡近似とそれを実現するアルゴリズム (2009)
  • Author(s): 二宮 祥一
  • Organizer: 計算による数理科学の展開2009(研究集会)
  • Place of Presentation: 神戸大学理学部
• [Presentation] A higher-order weak approximation method of SDEs and the Runge-Kutta method. (2009)
  • Author(s): 二宮 祥一
  • Organizer: Workshop “ Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance"
  • Place of Presentation: Wolfgang Pauli Institute, Vienna
• [Presentation] Malliavin calculus and Computational Fimamce (2008)
  • Author(s): S. Kusuoka
  • Organizer: Seoul-Tokyo Conference
  • Place of Presentation: KIAS
  • Year and Date: 2008-11-21
• [Presentation] Malliavin calculus, Computational Fimamce (2008)
  • Author(s): S. Kusuoka
  • Organizer: Symposium in Honor of Kiyosi Ito : Stochastic Analysis and Its Impact in Mathematics and Science
  • Place of Presentation: Institute of Mathematical Sciences National Univ. Singapore
  • Year and Date: 2008-07-10
• [Presentation] Malliavin calculus, Computational Fimamce (2008)
  • Author(s): S. Kusuoka
  • Organizer: Minisymposium on stochastic analysis in the occasion of the award the Degree of a Doctor Honoris Causa to Professor Paul Malliavin
  • Place of Presentation: the Faculty of Mathematics and Natural Sciences of the University of Bonn
  • Year and Date: 2008-04-19
• [Presentation] 楠岡近似のアルゴリズムについて (2008)
  • Author(s): 二宮祥一
  • Organizer: ファイナンスのための数理ワークショップ於早稲田大学理工学部
  • Place of Presentation: 東京
  • Year and Date: 2008-04-04
• [Presentation] Kusuoka Scheme : A new weak approximation methods of diffusion processes (2008)
  • Author(s): 二宮祥一
  • Organizer: 大阪大学金融保険教育研究センター主催中之島ワークショップ「金融工学・数理計量ファイナンスの諸問題2008」於大阪大学中之島センター
  • Place of Presentation: 大阪
• [Presentation] On the algorithms for the Kusuoka scheme (2006)
  • Author(s): Syoiti Ninomiya
  • Organizer: The International Conference on Mathematical Finance and its Applications
  • Place of Presentation: Kanazawa Univ., Kanazawa, Japan
• [Presentation] A new weak approximation scheme of stochastic differential equations by using the Runge-Kutta method (2006)
  • Author(s): Shigeo Kusuoka, Mariko Ninomiya, Syoiti Ninomiya
  • Organizer: Bachelier Finance Society 2006 4th World Congress Tokyo
  • Place of Presentation: Tokyo