On uniqueness of solutions to stochastic equations: a counter-example
Engelbert, H. J.
Ann. Probab., Tome 30 (2002) no. 1, p. 1039-1043 / Harvested from Project Euclid
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.
Publié le : 2002-07-14
Classification:  Stochastic equations,  continuous local martingales,  uniqueness in law,  60H10,  60G44
@article{1029867120,
     author = {Engelbert, H. J.},
     title = {On uniqueness of solutions to stochastic equations: a counter-example},
     journal = {Ann. Probab.},
     volume = {30},
     number = {1},
     year = {2002},
     pages = { 1039-1043},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1029867120}
}
Engelbert, H. J. On uniqueness of solutions to stochastic equations: a counter-example. Ann. Probab., Tome 30 (2002) no. 1, pp.  1039-1043. http://gdmltest.u-ga.fr/item/1029867120/