2011 Fiscal Year Final Research Report
Numerical Analysis of Jump-Models and Applications of Malliavin Calculus in Finance
| Project/Area Number |
21340024
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ritsumeikan University (2011)
Osaka University (2009-2010) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
AKAHORI Jiro 立命館大学, 理工学部, 教授 (50309100)
NAGAI Hideo 大阪大学, 基礎工学研究科, 教授 (70110848)
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| Co-Investigator(Renkei-kenkyūsha) |
AIDA Shigeki 東北大学, 理学部, 教授 (90222455)
KUSUOKA Sigeo 東京大学, 数理(科)学研究科(研究院), 教授 (00114463)
NINOMIYA Shoichi 東京工業大学, 大学院・イノベーションマネジメント研究科, 教授 (70313377)
KAWAI Reiichirou University of Leicester, Department of Mathematics, Lecturer (20464258)
TAKEUCHI Atsushi 大阪市立大学, 理学(系)研究科(研究院), 准教授 (30336755)
YAMAZATO Makoto 琉球大学, 理学部, 教授 (00015900)
YASUDA Kazuhiro 法政大学, 理工学部, 助教 (80509638)
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| Project Period (FY) |
2009 – 2011
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| Keywords | ミュレーション / 確率微分方程式 / ジャンプ型モデル / Malliavin解析 / 確率変数 / リスク / 誤差評価 / 非線形 |
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| Research Abstract |
In the present project, we applied Malliavin Calculus and operator decomposition techniques to study various problems in applied mathematics. In particular, we provided new simulation techniques for the approximation of solutions of stochastic differential equations driven by jumps. We studied the mathematical properties of these methods and proved that they provide a more accurate and fast method in comparison with past methods of simulation. This type of equation is also used as a model in Mathematical Finance. In this area we provided various formulas for the so called Greeks which measure risk in financial products. Through the use of the infinite dimensional integration by parts formula, we provide various alternatives and we also provided simulations to show the applicability of the results. We also obtain a lower bound estimate for Asian type random variables. We are thinking of applying these results to various problems in filtering.
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