Fractal and Stochastic Process

分形和随机过程

• 批准号: 09440086
• 负责人: KAMAE Teturo
• 金额: $ 4.74万
• 依托单位: Osaka City University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-09440086/
• 关键词: fractal; weighted substitution; colored tiling; homogeneous cocycle; self-similar process; 0-entropy; deterministic Brownian motion; 自己相似確率過程; O-エントロピー; 0-エントロピー

In this research, we studied among all a deterministic version of the Ito calculus.Deterministic Brownian motions are stochastic processes with noncorrelated, stationary and strictly ergodic increments having 0-entropy and 0-expectation. The self-similarity of order 1/2 follows from these properties. Such processes have a lot of variety and have different properties. It is not the case of the Brownian motion where the process is characterized as a process with stationary and independent increments with 0-expectation and standard variance.Among the deterministic Brownian motions, the simplest one is the N-process (N_t ; t∈R). We consider a process Y_t=H (N_t, t), where the function H(x, s) is twice continuously differentible in x and once continuously differentible in s and H_x(x, s)>0. The function H is consisered completely unknown except for these properties. We want to predict the value Y^c from the observation Y_J : ={Y_t ; t∈J}, where J=[a, b] and a<b< c. We proved that there exists a estimator Y_c such that<<numerical formula>>as c↓b with the following C (b) as the constant in O ( ) :<<numerical formula>>

在这项研究中，我们研究了伊藤微积分的确定性版本。确定性布朗运动是具有非相关、平稳和严格遍历增量的随机过程，具有 0 熵和 0 期望。1/2 阶的自相似性来自于。这些性质有很多变化，并且具有不同的性质，而布朗运动的过程则不是静止且独立的。增量为 0 期望和标准方差。在确定性布朗运动中，最简单的是 N 过程 (N_t ; t∈R)，我们考虑一个过程 Y_t=H (N_t, t)，其中函数 H(x)。 , s) 在 x 中连续可微，并且在 s 中连续可微，并且 H_x(x, s)>0 除了我们想要的这些属性之外，函数 H 被认为是完全未知的。根据观测值 Y_J 预测值 Y^c ；={Y_t t∈J}，其中 J=[a, b] 且 a<b< c 我们证明存在一个估计量 Y_c 使得<<数值公式> >as c↓b 并以下面的 C(b) 作为 O ( ) 中的常数：<<数值公式>>

项目成果

T.Kamae,Zhi-yin-Wen-Jun-ichi Tamura: "Hankel determinants for the Fibonacci word and Pade' approximation" Acta Arith.(to appear).

T.Kamae，Zhi-yin-Wen-Jun-ichi Tamura：“斐波那契字和 Pade 近似的 Hankel 行列式”Acta Arith。（即将出现）。

T.Kamae,J.M.Deshou Ileus J-M.Allouche,T.Koyanagi: "Automata, algebraicity and distribution of sequences of powers"Ann.Inst.Fourien. (印刷中).

T.Kamae、J.M.Deshou Ileus J-M.Allouche、T.Koyanagi：“自动机、代数性和幂序列的分布”Ann.Inst.Fourien（正在出版）。

N.Gjmi & T.Kamae: "Cobounday on colored tiling space as Rauzy fractal"Indogationes Mathematicae. 10-3. 407-421 (1999)

吉米

T.Kamae: "Linear expansions, sfrictly agodic homogeneous cocycles-"Israel J.Moth.. 106. 313-337 (1998)

T.Kamae：“线性展开式，严格的无极齐次余循环 -”Israel J.Moth.. 106. 313-337 (1998)

T.kamae,M.Keane: "A simple proof of the ratio ergodic theorem" Osaka J.Math.34. 653-657 (1997)

T.kamae，M.Keane：“比率遍历定理的简单证明”Osaka J.Math.34。