On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 483-497
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We prove that the uniqueness in law for an SDE \begin{equation} dX_t^i=b_t^i(X)\,dt+\sum_{j=1}^m\sigma_t^{ij}(X)\,dB_t^j, \qquad X_0^i=x^i,\quad i=1,\dots,n,\quad \tag{1} \end{equation} implies the uniqueness of the joint distribution of a pair $(X,B)$. Moreover, we prove that the uniqueness in law for (1), together with the strong existence, guarantees the pathwise uniqueness. This result is somehow “dual” to the theorem of Yamada and Watanabe.
Keywords:
stochastic differential equations, weak solutions, strong solutions, uniqueness in law, pathwise uniqueness, theorem of Yamada and Watanabe.
@article{TVP_2001_46_3_a4,
author = {A. S. Cherny},
title = {On the {Uniqueness} in {Law} and the {Pathwise} {Uniqueness} for {Stochastic} {Differential} {Equations}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {483--497},
year = {2001},
volume = {46},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a4/}
}
A. S. Cherny. On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 483-497. http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a4/