Decomposition of order statistics of semimartingales using local times

TL;DR

This paper extends the decomposition of order statistics for semimartingales to include discontinuous cases and generalizes existing formulas for local times of ranked processes, simplifying previous approaches.

Contribution

It introduces a more general semimartingale decomposition for non-continuous processes and generalizes key local time formulas, using a simpler method.

Findings

Derived a general decomposition for semimartingales with jumps.

Generalized formulas for local times of ranked processes.

Simplified the approach to semimartingale decomposition.

Abstract

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