2004 Fiscal Year Final Research Report Summary
Research on diffusion processes and fuzzy set valued random variables
| Project/Area Number |
15540127
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Saga University |
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Principal Investigator |
OGURA Yukio Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (00037847)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Hiroyuki Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (00190538)
SHIOYA Takashi Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
TOMISAKI Matsuyo Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (50093977)
MITOMA Itaru Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (40112289)
HANDA Kenji Saga University, Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (10238214)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Fuzzy set-valued random variable / central limit theorem / large deviation principle / Skorohod topology / Brownian motion / Markovian property / Levy's theorem / Pitman's theorem |
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| Research Abstract |
Study of random variables taking values in general spaces might be an important theme both for theoretical and applied mathematics. One of the objects of this research is to study limit theorems for fuzzy sets-valued random variables. It is worth to note that the target space looses separability with some topologies. One of the results of this research is having noticed that laws of large numbers, central limit theorems and martingale convergence theorems hold with respect to the uniform topology with which the fuzzy set space is not separable. We used the method exploiting monotone property and that reducing to the theory of empirical distribution by proving the integrability of the entropy in the procedure to let the mesh smaller.
Although the large deviation principles are more sensitive to the separability, we obtained Cramer type large deviation principles for as far as the topology induced by Levy's metric, and also for Skorohod topology and uniform topology with a little strong assumptions. We gave an explicit example which satisfies our assumptions. This seems to be a counter example to a result in a preprint appearing in an internet. We also obtained Sanov type large deviation principles under a natural assumption. Although we can compute rate functions explicitly only in simple cases, one of them is a relative entropy of two measures.
Also, with the investigator H.Matsumoto, we obtained that a cM-X process is Markov only when c=0,1,and 2,where X is a one-dimensional Brownnian motion with constant drift and M is its maximum process. This is another part of Levy's theorem(for c=1) and Pitman's theorem (for c=2).
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Research Products
(31 results)
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[Journal Article] Stochastic Holonomy2004
Author(s)
I.Mitoma
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Journal Title
Recent Developments in Stochastic Analysis and Related Topics (Sergio Albeverio et al.(eds.))(World Scientific)
Pages: 370-383
Description
「研究成果報告書概要(和文)」より
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[Journal Article] Stochastic Holonomy2004
Author(s)
I.Mitoma
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Journal Title
Recent Developments in Stochastic Analysis and Related Topics(Sergio Albeverio et al.(eds.)(World Scientific)
Pages: 370-383
Description
「研究成果報告書概要(欧文)」より
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