Comparison of path-independent functions of semimartingales

TL;DR

This paper extends the martingale comparison method to path-independent functions of general semimartingales, removing the need for Markovian assumptions and using non-Markovian Kolmogorov equations.

Contribution

It introduces a new comparison approach for semimartingales that does not rely on Markovian properties, broadening the applicability of the method.

Findings

Extended comparison results for non-Markovian semimartingales

Generalized Kolmogorov backward equations for non-Markovian processes

Established process comparison via Itô's formula on the same stochastic basis

Abstract

The martingale comparison method is extended to derive comparison results for path-independent functions for general semimartingales. Our approach allows to dismiss with the Markovian assumption on one of the processes made in previous literature. Main ingredients of the comparison method are extensions of the Kolmogorov backwards equation to the non-Markovian case. Putting the comparison processes on the same stochastic basis allows by means of It\^o's formula applied to the propagation operator to conclude the comparison of the processes from the comparison of the semimartingale characteristics.