On filtration enlargements and purely discontinuous martingales

Let M be a purely discontinuous martingale relative to a filtration . Given an arbitrary extension of the filtration , we will provide sufficient conditions for M to be a semimartingale relative to . Moreover we describe methods of how to find the Doob-Meyer decomposition with respect to the enlarged filtration. To this end we prove a new and more explicit version of the predictable representation property of Poisson random measures. Finally some concrete examples will show how the method developed may be applied.