2000 Fiscal Year Final Research Report Summary
Fractal and Stochastic Process
| Project/Area Number |
09440086
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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 | Osaka City University |
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Principal Investigator |
KAMAE Teturo Osaka City Univ., Prof., 理学部, 教授 (80047258)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAN Kenichi Osaka City Univ., Associated Prof, 理学部, 助教授 (70110856)
IMAYOSHI Yoichi Osaka City Univ., Prof., 理学部, 教授 (30091656)
KOMATSU Takasi Osaka City Univ., Prof., 理学部, 教授 (80047365)
伊達山 正人 大阪市立大学, 理学部, 講師 (10163718)
藤井 準二 大阪市立大学, 理学部, 講師 (60117968)
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| Project Period (FY) |
1997 – 2000
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| Keywords | fractal / weighted substitution / colored tiling / homogeneous cocycle / self-similar process / 0-entropy / deterministic Brownian motion |
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| Research Abstract |
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>>
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