  We derive equivalent conditions for the (local) absolute continuity of two
laws of semimartingales on random sets. Our result generalizes previous results
for classical semimartingales by replacing a strong uniqueness assumption by a
weaker uniqueness assumption. The main tool is a generalized Girsanov's
theorem, which relates laws of two possibly explosive semimartingales to a
candidate density process. Its proof is based on an extension theorem for
consistent families of probability measures. Moreover, we show that in a
one-dimensional It\^o-diffusion setting our result reproduces the known
deterministic characterizations for (local) absolute continuity. Finally, we
give a Khasminskii-type test for the absolute continuity of multi-dimensional
It\^o-diffusions and derive linear growth conditions for the martingale
property of stochastic exponentials.
