*Journées de Probabilités 2007*

**Stefan Ankirchner **

*How to find semimartingale decompositions relative to enlarged filtrations *

We will address the question under which conditions the semimartingale property of a stochastic process is preserved if the underlying filtration is enlarged. And we give an overview on old and new methods of how to derive semimartingale decompositions with respect to enlarged filtrations. We will see how to deal with arbitrary enlargements (incl. non-initial and non-progressive enlargements). Moreover we will discuss how to derive Doob-Meyer decompositions of pure jump martingales under enlarged filtrations. To this end we use a kind of Clark-Ocone formula for pure jump Levy martingales based on a difference operator introduced by Jean Picard.