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

In this paper, we derive a characterisation theorem for the path-independent property of the density of the Girsanov transformation for degenerated stochastic differential equations (SDEs), extending the characterisation theorem of Truman et al. (2012) for the non-degenerated SDEs. We further extend our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalises the corresponding characterisation theorem established in Qiao and Wu (submitted for publication).