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July 2002 On uniqueness of solutions to stochastic equations: a counter-example
H. J. Engelbert
Ann. Probab. 30(3): 1039-1043 (July 2002). DOI: 10.1214/aop/1029867120


We consider the one-dimensional stochastic equation \[ X_t=X_0+\int^t_0 b(X_s)\,dM_s \] where M is a continuous local martingale and b a measurable real function. Suppose that $b^{-2}$ is locally integrable. D. N. Hoover asserted that, on a saturated probability space, there exists a solution X of the above equation with $X_0=0$ having no occupation time in the zeros of b and, moreover, the pair (X, M) is unique in law for all such X. We will give an example which shows that the uniqueness assertion fails, in general.


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H. J. Engelbert. "On uniqueness of solutions to stochastic equations: a counter-example." Ann. Probab. 30 (3) 1039 - 1043, July 2002.


Published: July 2002
First available in Project Euclid: 20 August 2002

zbMATH: 1029.60045
MathSciNet: MR1920100
Digital Object Identifier: 10.1214/aop/1029867120

Primary: 60H10
Secondary: 60G44

Keywords: continuous local martingales , stochastic equations , uniqueness in law

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 3 • July 2002
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