On filtration enlargements and purely discontinuous martingales

Let MM be a purely discontinuous martingale relative to a filtration (Ft)(Ft). Given an arbitrary extension (Gt)(Gt) of the filtration (Ft)(Ft), we will provide sufficient conditions for MM to be a semimartingale relative to (Gt)(Gt). 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.