Study of fractional Brownian motion

分数布朗运动的研究

• 批准号: 10640107
• 负责人: KASAHARA Yuji
• 金额: $ 1.86万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640107/
• 关键词: fractional Brownian motion; Brownian motion; arc-sine law; diffusion process; self-similar process; self similarity; local time; tail probability; Gaussian process; Tauberian theorem; occupation time

1. Fractional Brownian motions are Gaussian processes having self-similarity. We studied the relationship between the Hearst index and the asymptotic behavior of the tail probabilities of their local times. In relation to this problem we studied the order of infinitesimal of the determinant of the covariance matrix as the dimension goes to infinity. We proved that it decreases exponentially and we found the relation between the exponent and the Hearst index. We also generalized the above results for more general Gaussian processes.2. When we studied the above problem we noticed that Tauberian theorems of exponential types are essential, and we obtained some useful theorems on this subject. As an application we studied the distribution function of the sums of independent random variables which are positive and identically distributed.3. We studied on some properties of self-similar processes.4. It is well known that the amount of time that a Brownian motion spends on the half line obeys the arc-sine law. We tried to find similar results for fractional Brownian motions but failed. Instead, however, we obtained an interesting result for linear diffusions : We found a relation between the so-called speed measure of the diffusion and the asymptotic behavior of the occupation time on the half line.

1. 分数布朗运动是具有自相似性的高斯过程。我们研究了赫斯特指数与其本地时间尾部概率的渐近行为之间的关系。关于这个问题，我们研究了当维数趋于无穷大时协方差矩阵行列式的无穷小阶。我们证明了它呈指数递减，并找到了指数与赫斯特指数之间的关系。我们还将上述结果推广到更一般的高斯过程。2．当我们研究上述问题时，我们注意到指数类型的陶伯定理是必不可少的，并且我们在这个问题上得到了一些有用的定理。作为一个应用，我们研究了正同分布的独立随机变量之和的分布函数。 3．研究了自相似过程的一些性质； 4．众所周知，布朗运动在半线上花费的时间遵循反正弦定律。我们试图为分数布朗运动找到类似的结果，但失败了。然而，相反，我们获得了线性扩散的有趣结果：我们发现了所谓的扩散速度测量与半线上占据时间的渐近行为之间的关系。

项目成果

M.Maejima et al.: "Operator semi-selfdecomposability, (C,Q)-decomposability and related nested classes"Tokyo J. Math.. 22. 473-509 (1999)

M.Maejima 等人：“算子半自分解性、(C,Q)-分解性和相关嵌套类”Tokyo J. Math.. 22. 473-509 (1999)

M.Maejima et al.: "Exponents of semi-selfsimilar processes"Yokohama Math. J.. 47. 93-102 (1999)

M.Maejima 等：“半自相似过程的指数”横滨数学。

N.Kosugi: "Tauberian theorem of exponential type and its application to multiple convolution"J. Math. Kyoto Univ.. 39. 331-346 (1999)

N.Kosugi：“指数型陶伯定理及其在多重卷积中的应用”J.

N.Kosugi: "Functional limit theorem for occupation times of Gaussian processes -non-critical case" Osaka J.Math.(印刷中). (1999)

N.Kosugi：“高斯过程占用时间的函数极限定理 - 非关键情况”Osaka J.Math（出版中）。

Y.Kasahara et al.: "On tail probability of local times of Gaussian processes"Stoch. Proc. Appl.. 82. 15-21 (1999)

Y.Kasahara 等人：“高斯过程局部时间的尾部概率”Stoch。