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 Electronic Communications in Probability > Vol. 6 (2001) > Paper 6 open journal systems 


A 2-Dimensional SDE Whose Solutions are Not Unique

Jan M. Swart, University of Erlangen-Nuremberg


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)= deltaijs(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-by-m matrices. We construct a 2-by-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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Pages: 67-71

Published on: July 12, 2001


Bibliography
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  2. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, New York, 1986. Math. Reviews number 88a:60130
  3. J. M. Swart, Pathwise uniqueness for a SDE with non-Lipschitz coefficients. preprint 2000. Math. Reviews number not available.
  4. T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 11:155-167, 1971. Math. Reviews number 43 #4150
  5. T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ., 11:553-563, 1971. Math. Reviews number 44 #6071
















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Electronic Communications in Probability. ISSN: 1083-589X