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.