2005 Fiscal Year Final Research Report Summary
Study on semi-Levy processes and semi-selfsimilar processes
| Project/Area Number |
15340036
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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 | KEIO UNIVERSITY |
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Principal Investigator |
MAEJIMA Makoto Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90051846)
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| Co-Investigator(Kenkyū-buntansha) |
NAKADA Hitoshi Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (40118980)
TANI Atusi Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90118969)
TAMURA Yozo Keio Univ., Faculty of Sci.and Tech., Associate Professor, 理工学部, 助教授 (50171905)
SATO Ken-iti Nagoya University, Emeritus, Professor, 名誉教授 (60015500)
WATANABE Toshiro The University of Aizu, Center of Mathematical Sciences, Lecturer, 総合数理科学センター, 専任講師 (50254115)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Levy process / semi-Levy process / infinitely divisible distribution / stochastic integral characterization / semi-selfsimilar process / additive parocess |
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| Research Abstract |
This year was the last year of the three year project, and we spent most time for the study of stochastic integral characterizations of several subclasses of infinitely divisible disitirutions in connection with the purpose of the whole project. As a result, we succeeded in stochastic integral characterization of the class of type G distributions and its decreasing sequence of subclasses, which was one of the aims at the beginning of the academic year of 2005. Furthermore, we found a new subclass of type G and seldecomposable distributions which has very simple form of Levy measures in the Levy-Khintchine representation of the characteristic function of infinitely divisible distribution. Maejima gave a talk on this result together with more other related results at the seminar of Munich Technical University, Germany, November 2005, and the talk was very much appreciated. Although we were not able to get a big success on semi-Levy processes, Maejima suggested in the seminar the idea of
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replacing Levy process in the stochastic integral by semi-Levy process, and this idea was well accepted by the audience, and Maejima felt a wave of a new international joint research on this topic. These works were mainly done by Maejima and Sato, Maejima, his student and Jan Rosinski at University of Tenessee.
Another topic which was mentioned at the beginning of this academic year was a new topic, which is generalized Ornstein-Uhlenbeck processes. We investigated the infinite divisibility of the limiting distributions of generalized Ornstein-Uhlenbeck processes and we found that under a mild condition they are not only infinitely divisible but also seldecomposable. Also several interesting examples were constructed. These results were presented by Maejima at an International Conference in Mexico, March 2006.
Based on the results obtained during the three year project, we can see the route of the study of Semi-Levy processes and more precise classification of the class of infinitely divisible distributions. Besides those, Maejima found, together with Victor Perez-Abrue at Maxico, examples of random matrices whose determinants are infinitely divisible. These examples are interesting because it is known that the original random matrices are not infinitely divisible. A lot of interesting unsolved problems around the infinitely divisibility remain. Less
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