Gauge Theory of Stochastic Processes
| Project/Area Number |
15K17562
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Fukuoka University (2017-2018)
Ritsumeikan University (2015-2016) |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 超関数に対する伊藤の公式 / Wiener汎関数の正則化 / Malliavin解析 / 双対確率流 / Loewner-Kufarev方程式 / Grunsky係数 / tau関数 / 定常Gauss過程 / Level Crossings / Signature / Witt代数 / Skorokhod積分 / smoothing effect / 離散時間確率流 / 局所時間 / 伊藤の公式 |
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| Outline of Final Research Achievements |
1. It is proven that the scheme approximating an irregular function of a Levy process by discretizing the Clark-Ocone formula with respect to time has the strong-convergence rate given by half of the Sobolev index of the functional (Joint with Nien-Lin Liu, Azmi Makhlouf and Takwa Saidaoui). 2. We have introduced a class which we call controlled Loewner-Kufarev equation. We embedded the solution to the Sato-Segal-Wilson Grassmannian accoding to the procudure of Krichever, and then we described the time-evolution in the Grassmannian and obtained an explicit formula describing the associated Grunsky coefficients (Joint with Roland Friedrich). 3. We proved that integration with respect to time of the function of a solution to a stochastic differential equation (or smooth stationary Gaussian process and its derivative) raise the Sobolev index by 1 (1/2 respectively) than that of the integrand (Joint with Yoshihiro Ryu and Kratz Marie).
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| Academic Significance and Societal Importance of the Research Achievements |
1はオプションのペイオフ関数の近似において、その関数が滑らかでないときにも、その少ない滑らかさに応じて近似の精度を具体的に保証するという意味で実務的にも有意義である。2はこれまで確率論の文脈においては統計モデルのみに用いられていたLoewner方程式の、無限可積分系における立ち位置を示唆しているように見える、という意味で分野を横断して意義がある。3や4は機械学習の中でもGauss過程回帰モデルの文脈においてlength scaleと呼ばれるハイパーパラメータを、Rice公式とMC法を用いて推定する際に、その精度を保証することに繋がるため、このモデルを実装する社会の様々な文脈において有用である。
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Report
(5 results)
Research Products
(22 results)
