A 2-Dimensional SDE Whose Solutions are Not Unique

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.