Open Access
2001 A 2-Dimensional SDE Whose Solutions are Not Unique
Jan Swart
Author Affiliations +
Electron. Commun. Probab. 6: 67-71 (2001). DOI: 10.1214/ECP.v6-1035

Abstract

In 1971, Yamada and Watanabe showed that pathwise uniqueness holds for the SDE dX=σ(X)dB when sigma takes values in the n-by-m matrices and satisfies |σ(x)σ(y)|<|xy|log(1/|xy|)1/2. When n=m=2 and σ is of the form σij(x)=δijs(x), they showed that this condition can be relaxed to |σ(x)σ(y)|<|xy|log(1/|xy|), leaving open the question whether this is true for general 2×m matrices. We construct a 2×1 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.

Citation

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Jan Swart. "A 2-Dimensional SDE Whose Solutions are Not Unique." Electron. Commun. Probab. 6 67 - 71, 2001. https://doi.org/10.1214/ECP.v6-1035

Information

Accepted: 12 July 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 0988.60059
MathSciNet: MR1846542
Digital Object Identifier: 10.1214/ECP.v6-1035

Keywords: diffusion process , pathwise uniqueness / strong uniqueness , Stochastic differential equation

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