Electronic Communications in Probability

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

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)= delta_ijs(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.

Full text: PDF

Pages: 67-71

Published on: July 12, 2001

Bibliography

M. T. Barlow, One dimensional stochastic differential equations with no strong solution, J. London Math. Soc., 26:335-347, 1982. Math. Reviews number 84d:60083
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, New York, 1986. Math. Reviews number 88a:60130
J. M. Swart, Pathwise uniqueness for a SDE with non-Lipschitz coefficients. preprint 2000. Math. Reviews number not available.
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
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