Comparison of Markov processes by the martingale comparison method

TL;DR

This paper extends the martingale comparison method to general Markov processes, providing a new approach for comparing Markov processes using supermartingale properties and martingale problem characterizations.

Contribution

It introduces a novel martingale comparison technique for Markov processes, broadening the applicability beyond evolution system-based methods.

Findings

Provides a new comparison result for Markov processes

Uses martingale problem characterization for supermartingale derivation

Offers regularity conditions different from existing evolution system approaches

Abstract

Comparison results for Markov processes w.r.t. function class induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. In this paper this property is achieved using in an essential way the characterization of Markov processes by the martingale problem. As a result the martingale comparison method gives a comparison result for Markov processes under a general alternative set of regularity conditions compared to the evolution system approach.