  In this paper, we derive a characterization theorem for the path-independent
property of the density of the Girsanov transformation for {\it degenerated}
stochastic differential equations (SDEs), extending the characterization
theorem of \cite{twwy} for the non-degenerated SDEs. We further extends our
consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion
coefficients, which generalizes the corresponding characterization theorem
established in \cite{hqwu}.
