Semimartingales and shrinkage of filtration

We consider a complete probability space (Ω, F, P), which is endowed with two filtrations, G and F , assumed to satisfy the usual conditions and such that F ⊂ G. On this probability space we consider a real valued G -semimartingale X. The purpose of this work is to study the following two problems: A. If X is F -adapted, compute the F -semimartingale characteristics of X in terms of the G -semimartingale characteristics of X. B. If X is a special G -semimartingale but not F -adapted, compute the F -semimartingale characteristics of the F -optional projection of X in terms of the G -canonical decomposition and the G -semimartingale characteristics of X. In this paper problem B is solved under the assumption that the filtration F is immersed in G . Beyond the obvious mathematical interest, our study is motivated by important practical applications in areas such as finance and insurance (cf. Structured Dependence Between Stochastic Processes (2020) Cambridge Univ. Press).