Decomposition of order statistics of semimartingales using local times

In a recent work \cite{BG}, given a collection of continuous semimartingales, authors derive a semimartingale decomposition from the corresponding ranked processes in the case that the ranked processes can meet more than two original processes at the same time. This has led to a more general decomposition of ranked processes. In this paper, we derive a more general result for semimartingales (not necessarily continuous) using a simpler approach. Furthermore, we also give a generalization of Ouknine \cite{O1, O2} and Yan's \cite{Y1} formula for local times of ranked processes