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A 2-Dimensional SDE Whose Solutions are Not Unique


 
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1. Title Title of document A 2-Dimensional SDE Whose Solutions are Not Unique
 
2. Creator Author's name, affiliation, country Jan M. Swart; University of Erlangen-Nuremberg
 
3. Subject Discipline(s) Mathematics
 
3. Subject Keyword(s) Stochastic differential equation, pathwise uniqueness / strong uniqueness, diffusion process.
 
4. Description Abstract In 1971, Yamada and Watanabe showed that pathwise uniqueness holds for the SDE $dX= \sigma (X)dB$ when sigma takes values in the n-by-m matrices and satisfies $|\sigma (x)- \sigma (y)| < |x-y|\log(1/|x-y|)^{1/2}$. When $n=m=2$ and $\sigma$ is of the form $\sigma _{ij}(x)= \delta_{ij}s(x)$, they showed that this condition can be relaxed to $| \sigma(x)-\sigma(y)| < |x-y|\log(1/|x-y|)$, leaving open the question whether this is true for general $ 2\times m$ matrices. We construct a $2\times 1$ matrix-valued function which negatively answers this question. The construction demonstrates an unexpected effect, namely, that fluctuations in the radial direction may stabilize a particle in the origin.
 
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7. Date (YYYY-MM-DD) 2001-07-12
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1035
 
10. Identifier Digital Object Identifier 10.1214/ECP.v6-1035
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 6
 
12. Language English=en en
 
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