On the canonical extension of stochastic differential equations based on semimartingales with spatial parameters
Project/Area Number  09640269 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Hyogo University of Teacher Education 
Principal Investigator 
FUJIWARA Tsukasa Hyogo University of Teacher Education, 学校教育学部, 助教授 (30199385)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  SDE / martingale / stochastic flow / 確率微分方程式 / マルチンゲール / 確率流 / セミマルチンゲル / 微分同相群 
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 ddimensional 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.

Report
(4results)
Research Output
(6results)