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In this paper, we derive a characterization theorem for the path-independentproperty of the density of the Girsanov transformation for {\it degenerated}stochastic differential equations (SDEs), extending the characterizationtheorem of \cite{twwy} for the non-degenerated SDEs. We further extends ourconsideration to non-Lipschitz SDEs with jumps and with degenerated diffusioncoefficients, which generalizes the corresponding characterization theoremestablished in \cite{hqwu}.

Characterizing the path-independent property of the Girsanov density for degenerated stochastic differential equations