ゲーム論的確率論の予測理論への応用

• Project/Area Number: 18650070
• Research Category: Grant-in-Aid for Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Statistical science
• Research Institution: The University of Tokyo
• Principal Investigator: 竹村 彰通 The University of Tokyo, 大学院・情報理工学系研究科, 教授 (10171670)
• Co-Investigator(Kenkyū-buntansha): 駒木 文保 東京大学, 大学院・情報理工学系研究科, 准教授 (70242039) ; 清 智也 東京大学, 大学院・情報理工学系研究科, 助教 (20401242)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,100,000 (Direct Cost: ¥3,100,000); Fiscal Year 2008: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 大数の強法則 / 重複対数の法則 / 中心極限定理

Research Abstract

Vladimir VovkとGlenn Shafer によるゲーム論的確率論は以下の著書でその基礎が与えられた.
Glenn Shafer and Vladimir Vovk. Probability and Finance:It's Only a Game! Wiley, New York, 2001
そこでは,Skepticとよばれる賭をする人と,Realityとよばれる賭の結果を定める人の,二人のプレーヤーの間のゲームを設定することにより,ゲームの結果として確率が定まることが示されている.注目すべきは,測度論無しに,大数の強法則,中心極限定理,重複対数の法則,さらに数理ファイナンスにおける価格付けの諸公式,などが証明される点にある.竹村は,竹内啓,公文雅之との共同研究を通じて,ゲーム論的確率論に関する新たな結果を得ている.これらはtechnical reportとして発表されていたが,研究発表に示すように国際雑誌に刊行の段階となっている.
また竹村はShafer氏およびVovk氏とも共同研究を進めており,以下の研究成果を得た."The game-theoretic martingales behind the zero-one laws", by Akimichi Takemura, Vladimir Vovk and Glenn Shafer. Techinical Report METR 08-18, March 2008.この研究では,測度論の諸仮定をおくことなく,コルモゴロフの0-1法則などの0-1法則の一般的な形をゲーム論的枠組で示している.

Research Products

• [Journal Article] Capital process and optimality properties of a Bayesian skeptic in coin-tossing games (2008)
  • Author(s): Masayuki Kumon, A. Takemura and Kei Takeuchi
  • Journal Title: Stochastic Analysis and Applications 26 Pages: 1161-1180
  • Peer Reviewed
• [Journal Article] Implications of contrarian and one-sided strategies for the fair-coin game. (2008)
  • Author(s): Yasunori Horikoshi and A. Takemura.
  • Journal Title: Stochastic Processes and their Applications 118 Pages: 2125-2142
  • Peer Reviewed
• [Journal Article] On a simple strategy weakly forcing the storong law of large numbers in the bounded forecasting game. (2008)
  • Author(s): Masayuki Kumon and A. Takemura
  • Journal Title: Annals of the Institute of Statistical Mathematics, 60 Pages: 801-812
  • Peer Reviewed
• [Journal Article] Game-theoretic versions of strong law of large numbers for unbounded variables (2007)
  • Author(s): Masayuki Kumon, Akimichi Takemura and Kei Takeuchi
  • Journal Title: Stochastics 79 Pages: 449-468
  • Peer Reviewed
• [Journal Article] Game-theoretic derivation of discrete distributions and discrete pricing formulas (2007)
  • Author(s): Akimichi Takemura and Taiji Suzuki
  • Journal Title: Journal of the Japan Statistical Society 37 Pages: 87-104
  • NAID: 110006317398
  • Peer Reviewed
• [Journal Article] On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game (2007)
  • Author(s): Masayuki Kumon, Akimichi Takemura
  • Journal Title: Annals of the Institute of Statistical Mathematics, DOI 10. 1007/s10463-007-0125-5
  • NAID: 40016356934
• [Journal Article] Game-theoretic versions of strong law of large numbers for unbounded variables (2007)
  • Author(s): Masayuki Kumon, Akimichi Takemura, Kei Takeuchi
  • Journal Title: Stochastics (to appear)
• [Journal Article] Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games (2007)
  • Author(s): Masayuki Kumon, Akimichi Takemura, Kei Takeuchi
  • Journal Title: Journal of Stochastic Analysis and Applications (to appear)