New higher order discretization method with Malliavin calculus
| Project/Area Number |
16K13773
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Hitotsubashi University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | マリアバン解析 / 確率微分方程式 / 弱近似 / 確率微分方程式の弱近似 / 高次離散化法 |
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| Outline of Final Research Achievements |
In the project, we studied weak approximation schemes for stochastic differential equations driven by Brownian motions. As a result, we obtained a new higher order algorithm, the Malliavin Monte Carlo, which is constructed by some polynomials of Brownian motions through the Malliavin calculus. Not only for the standard weak approximation problem, a higher order discretization scheme for the differentiations of the solutions to partial differential equations is also obtained. We further provide another type of weak approximation using a Markov chain approximation. These results are published in major academic journals in the areas of applied mathematics such as numerical analysis, statistics and finance.
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| Academic Significance and Societal Importance of the Research Achievements |
自然科学・社会科学においてランダムな微分方程式によるモデルは頻繁に現れ、さらにそれらのモデルの対象の平均的な振る舞いや確率計算は明らかにしたい現象を解析する上で本質的に重要である。モデルが単純であれば解析を行うことは簡単であるが、一般にはモデルは複雑になりうる。それは自然科学・社会科学の問題が単純でなく複雑だからである。本研究の成果はこのような問題に寄与しうる。従来のシミュレーション法は収束が遅いが、本研究の方法ではマリアバン解析を用いた補正を加えるので収束が非常に速くなり、また感覚的にも分かりやすい。物理現象の解析、機械学習、金融商品の数値計算やリスク管理など多くの分野への応用が期待される。
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Report
(4 results)
Research Products
(13 results)
