On the construction of conditional probability densities in the Brownian and compound Poisson filtrations (Q6617084)

The authors construct supermartingales with values in \([0,1]\) as solutions to stochastic differential equations on the given Brownian reference filtration. Following the work [\textit{M. Jeanblanc} and \textit{S. Song}, Stochastic Processes Appl. 121, No. 8, 1678--1704 (2011; Zbl 1298.91176)], they construct associated random times on appropriate extended probability spaces. Using a multiplicative decomposition of the supermartingales constructed at the first step, the authors point out the families of the conditional probability densities of the random times.\N\NAn analogous program is also carried out in the case of a compound Poisson reference filtration.