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The martingale comparison method for Markov processes

Published online by Cambridge University Press:  25 February 2021

Benedikt Köpfer*
Affiliation:
University of Freiburg
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
*Postal address: Department of Mathematical Stochastics, University of Freiburg, Freiburg, Germany.
*Postal address: Department of Mathematical Stochastics, University of Freiburg, Freiburg, Germany.

Abstract

Comparison results for Markov processes with respect to 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. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

Type
Research Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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