Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T10:56:50.388Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Comparison Result for Markov Processes

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Ludger Rüschendorf
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

Keywords

Association of Markov processesstochastic monotonicityinfinitesimal generatorstochastic ordering

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 279 - 286
Copyright
Copyright © Applied Probability Trust 2008 

References

Bäuerle, N., Müller, A. and Blatter, A. (2008). Dependence properties and comparison results for Lévy processes. Math. Meth. Operat. Res. 67, 161–186.CrossRefGoogle Scholar
Bergenthum, J. and Rüschendorf, L. (2007). Comparison of semimartingales and Lévy processes. Ann. Prob. 35, 228–254.Google Scholar
Chen, M.-F. (2004). From Markov Chains to Non-equilibrium Particle Systems. World Scientific, Singapore.Google Scholar
Chen, M.-F. and Wang, F.-Y. (1993). On order preservation and positive correlations for multidimensional diffusion processes. Prob. Theory Relat. Fields 95, 421–428.Google Scholar
Christofides, T. C. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. J. Multivariate Anal. 88, 138–151.Google Scholar
Cox, T. (1984). An alternative proof of a correlation inequality of Harris. Ann. Prob. 12, 272–273.Google Scholar
Cox, T., Fleischmann, K. and Greven, A. (1996). Comparison of interacting diffusions and an application to their ergodic theory. Prob. Theory Relat. Fields 105, 513–528.Google Scholar
Daduna, H. and Szekli, R. (2006). Dependence ordering for Markov processes on partially ordered spaces. J. Appl. Prob. 43, 793–814.Google Scholar
Greven, A., Klenke, A. and Wakolbinger, A. (2002). Interacting diffusions in a random medium: comparison and long-time behaviour. Stoch. Process. Appl. 98, 23–41.Google Scholar
Harris, T. E. (1977). A correlation inequality for Markov processes in partially ordered state spaces. Anna. Prob. 5, 451–454.Google Scholar
Herbst, I. and Pitt, L. (1991). Diffusion equation techniques in stochastic monotonicity and positive correlations. Prob. Theory Relat. Fields 87, 275–312.Google Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingale (Lecture Notes Math. 714). Springer, Berlin.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
Massey, W. A. (1987). Stochastic ordering for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Pitt, L. (1982). Positively correlated normal variables are associated. Ann. Prob. 10, 496–499.Google Scholar
Rüschendorf, L. (1980). Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitsth. 54, 341–349.Google Scholar
Samorodnitsky, G. (1995). Association of infinitely divisible random vectors. Stoch. Process. Appl. 55, 45–55.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.Google Scholar