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マリアヴァン解析の数理ファイナンスへの応用の研究

Mariavan分析在数理金融中的应用研究

基本信息

批准号：
11874029
负责人：
谷口説男
金额：
$ 1.34万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Exploratory Research
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

マリアヴァン解析に基づくクラーク・オコーンの公式を用いて条件付き請求権をヘッジするポートフォリオを計算する手法の拡張について,とくに終端時刻においてピン留め条件を付加した条件付き確率のもとでの拡張について研究を行った.そのために確率微分方程式の解を終端時刻でピン留めする際の解の挙動について調べた.その結果,線形なドリフト項をもつ確率微分方程式の解にピン留め条件を付加すれば,その確率微分方程式の定める確率流が終端時刻において微分同相写像から定数写像への縮退を非常によい減衰のオーダーを持って起こすことを見いだした.これにより,ガウス型の確率微分方程式の解を終端時刻でピン留めする条件付き確率の下ではクラーク・オコーンの公式が拡張できることを証明した.この性質は一般の単連結リーマン多様体上のブラウン運動への拡張が期待でき,現在その拡張を継続研究中である.また,ピン留め確率測度の研究の一環として,時刻1で原点にピン留めする重み付きピン留めウィナー測度の下でレヴィの確率面積の特性関数について厳密表現を与えた.この結果は論文にまとめ発表した.以上の結果を踏まえ,12年11月末にパリのピエール・マリー=キュリー・デニス=デュデロ大学で開かれた確率論国際研究集会「日仏確率論研究会」において講演し,またその研究集会前後にマリアヴァン教授(フランス学士院会員),ヨール教授(パリ第6大学)との研究討論を行い,ピン留め測度の下での確率解析,マリアヴァン解析の応用についての知見を広めた.現在,これらの結果に基づき研究を継続中である.

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