Abstract
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.
Citation
H. J. Engelbert. "On uniqueness of solutions to stochastic equations: a counter-example." Ann. Probab. 30 (3) 1039 - 1043, July 2002. https://doi.org/10.1214/aop/1029867120
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