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 differ- ential equations (SDEs), extending the characterisation theorem of [13] for the non- degenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalises the corresponding characterisation theorem established in [10]