In this paper, we construct supermartingales valued in [0, 1] as solutions of an appropriate stochastic differential equation on a given reference filtration generated by either a Brownian motion or a compound Poisson process. Then, by means of the results contained in [19], it is possible to construct an associated random time on some extended probability space admitting such a given supermartingale as conditional survival process and we shall check that this construction (with a particular choice of supermartingale) implies that Jacod's equivalence hypothesis, that is, the existence of a family of strictly positive conditional probability densities for the random times with respect to the reference filtration, is satisfied. We use the components of the multiplicative decomposition of the constructed supermartingales to provide explicit expressions for the conditional probability densities of the random times on the Brownian and compound Poisson filtrations. * This research benefited from the support of the 'Chaire Marchés en Mutation', French Banking Federation and ILB, Labex ANR 11-LABX-0019.