On filtration enlargements and purely discontinuous martingales

Let M be a purely discontinuous martingale relative to a filtration ( F t ) . Given an arbitrary extension ( G t ) of the filtration ( F t ) , we will provide sufficient conditions for M to be a semimartingale relative to ( G t ) . 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.