On filtration enlargements and purely discontinuous martingales
The goal of the article under review is to know when a purely discontinuous martingale \(M_t\) under a filtration \({\mathcal F}_t\) remains a semimartingale after an enlargement of the filtration. If \({\mathcal G}_t\) is the enlarged filtration, a sufficient condition is given, and a new explicit formula for the Doob-Meyer decomposition of \(M_t\) under \({\mathcal G}_t\) is obtained. More precisely, the finite variation part of \(M_t\) is called the information drift of \(M_t\) with respect to \({\mathcal G}_t\), and the formula for this drift involves the predictable representation of the martingales \(E[1_A| {\mathcal F}_t]\). On the other hand, it is known that such a predictable representation can be obtained by means of a difference operator on the Poisson space, and this leads to another expression for the information drift. Finally, the method is applied to some concrete examples.
scientific article; zbMATH DE number 5348271
scientific article