On the lack of semimartingale property

TL;DR

This paper extends the understanding of when functions of semimartingales retain the semimartingale property, showing they must be differences of convex functions on intervals, and applies this to analyze the median process.

Contribution

It generalizes semimartingale function characterization to non-Markovian settings and demonstrates the median process is not a semimartingale.

Findings

Functions of semimartingales are differences of convex functions on intervals.

The median process is proven not to be a semimartingale.

Extension of semimartingale characterization to non-Markovian processes.

Abstract

In this work we extend the characterization of semimartingale functions in Cinlar et al. (1980) to the non-Markovian setting. We prove that if a function of a semimartingale remains a semimartingale, then under certain conditions the function must have intervals where it is a difference of two convex functions. Under suitable conditions this property also holds for random functions. As an application, we prove that the median process defined in Prokaj et al. (2011) is not a semimartingale. The same process appears also in Hu and Warren (2000) where the question of the semimartingale property is raised but not settled.