Enlargement of filtrations with random times for processes with jumps

We treat an extension of Jacod's theorem for initial enlargement of filtrations with respect to random times. In Jacod's theorem the main condition requires the absolute continuity of the conditional distribution of the random time with respect to a nonrandom measure. Examples appearing in the theory on insider trading require extensions of this theorem where the reference measure can be random. In this article we consider such an extension which leads to an extra term in the semimartingale decomposition in the enlarged filtration. Furthermore we consider a slightly modified enlargement which allows for the bounded variation part of the semimartingale decomposition to have finite moments depending on the modification considered. Various examples for Lévy processes are treated.