1998 Fiscal Year Final Research Report Summary
On the canonical extension of stochastic differential equations based on semimartingales with spatial parameters
| Project/Area Number |
09640269
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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 | Hyogo University of Teacher Education |
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Principal Investigator |
FUJIWARA Tsukasa Hyogo University of Teacher Education, 学校教育学部, 助教授 (30199385)
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| Project Period (FY) |
1997 – 1998
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| Keywords | SDE / martingale / stochastic flow |
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| Research Abstract |
In this research, a certain type of stochastic differential equations (SDE's) based on C(R^d, R^d) valued sernimartingales, which will be called to be canonical, are investigated and the following results are obtained. Here, C(Rd, Rd) denotes the space of continuous mappings from the d-dimensional Euclidean space Rd to itself.
(1) The definition of canonical integrals based on C(R^d, R^d) valued semimartingales as a class of stochastic integrals and the relationship to the Ito integrals and the Stratonovich integrals.
(2) The existence and uniqueness of the solutions of canonical SDE's.
(3) Homeomorphic and moreover diffeomorphic properties of the solutions.
(4) A fact that the inverse of the stochastic flow of diffeomorphisrns generated by a canonical SDE is represented as a system of solutions of the corresponding backward canonical SDE under some suitable conditions.
Through this research, it is shown that by their own structure canonical SDE's naturally generate several nice properties in relation to the theory of stochastic flows.
These results were reported at a symposium of the Mathematical Society of Japan in September 1998. Furthermore, the details will be published in Kyushu Journal of Mathematics as a pair of papers written by the head investigator and Professor Hiroshi Kunita.
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