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

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}.