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Comparison of time-inhomogeneous Markov processes

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  11 January 2017

Ludger Rüschendorf ,
Alexander Schnurr  and
Viktor Wolf
Ludger Rüschendorf*
Affiliation:
University of Freiburg
Alexander Schnurr*
Affiliation:
Siegen University and TU Dortmund
Viktor Wolf*
Affiliation:
University of Freiburg
*
* Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstraße 1, 79104 Freiburg, Germany.
** Postal address: Department of Mathematics, Siegen University, Walter-Flex-Straße 3, 57068 Siegen, Germany. Email address: schnurr@mathematik.uni-siegen.de
* Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstraße 1, 79104 Freiburg, Germany.
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Abstract

Comparison results are given for time-inhomogeneous Markov processes with respect to function classes with induced stochastic orderings. The main result states the comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments, and to Lévy-driven diffusion processes.

Keywords

Stochastic orderingMarkov processinfinitesimal generatorprocesses with independent incrementsevolution system

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J35: Transition functions, generators and resolvents 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1015 - 1044
Copyright
Copyright © Applied Probability Trust 2017 

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