On the optional and orthogonal decompositions of a class of semimartingales

Abstract

We consider a set \mathcal Q of probability measures, which are absolutely continuous with respect to the physical probability measure \mathbb{P} and at least one is equivalent to \mathbb{P} . We investigate necessary and sufficient conditions on \mathcal Q , under which any \mathcal Q -supermartingale can be decomposed into the sum of a local \mathcal Q -martingale and a decreasing process. We also provide an orthogonal decomposition of square integrable semimartingale as the orthogonal sum of a local \mathcal Q -martingale and a square integrable semimartingale. As one application, we state the orthogonal decomposition in an appropriate sense of the polar set of \mathcal Q . We generalise then the results of a previous article (2021), from finite probability space and discrete time case to general continuous time case.