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_{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-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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