On the construction of conditional probability densities

In this paper, we construct strictly positive conditional probability densities with respect to the given reference filtration in the two cases of a filtration generated by a Brownian motion and a (compound) Poisson process. Then, by means of the results contained in [23], it is possible to construct the associated random times on some extended probability space. Hence, Jacod's equivalence hypothesis, that is, the existence of strictly positive conditional densities for the random times with respect to the reference filtration, is obviously satisfied