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

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.