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On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations

A. S. Cherny

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Received: 18.05.2001

DOI: 10.4213/tvp3897

 English version: Theory of Probability and its Applications, 2002, 46:3, 406–419

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