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.
References
M. T. Barlow, One dimensional stochastic differential equations with no strong solution, J. London Math. Soc., 26:335-347, 1982. MR675177 0456.60062 10.1112/jlms/s2-26.2.335M. T. Barlow, One dimensional stochastic differential equations with no strong solution, J. London Math. Soc., 26:335-347, 1982. MR675177 0456.60062 10.1112/jlms/s2-26.2.335
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, New York, 1986. 0592.60049S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, New York, 1986. 0592.60049
J. M. Swart, Pathwise uniqueness for a SDE with non-Lipschitz coefficients. preprint 2000. 1058.60047 10.1016/S0304-4149(01)00140-5J. M. Swart, Pathwise uniqueness for a SDE with non-Lipschitz coefficients. preprint 2000. 1058.60047 10.1016/S0304-4149(01)00140-5
T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 11:155-167, 1971. 0236.60037 10.1215/kjm/1250523691 euclid.kjm/1250523691T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 11:155-167, 1971. 0236.60037 10.1215/kjm/1250523691 euclid.kjm/1250523691
T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ., 11:553-563, 1971. MR0288876 0229.60039 10.1215/kjm/1250523620 euclid.kjm/1250523620T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ., 11:553-563, 1971. MR0288876 0229.60039 10.1215/kjm/1250523620 euclid.kjm/1250523620