Comptes Rendus
no. 12
Probability Theory
Monotonicity and complete monotonicity for continuous-time Markov chains
[Monotonie et monotonie complète des chaînes de Markov à temps continu]

Présenté par : Jean-Pierre Kahane

Paolo Dai Pra1 ; Pierre-Yves Louis2 ; Ida Minelli1
1 Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy
2 Institut für Mathematik, Potsdam Universität, Am neuen Palais, 10 – Sans Souci, 14415 Potsdam, Germany
Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 965-970.

Nous étudions les notions de monotonie et de monotonie complète pour les processus de Markov (ou chaînes de Markov à temps continu) prenant leurs valeurs dans un espace partiellement ordonné. Ces deux notions ne sont pas équivalentes, comme c'est le cas lorsque le temps est discret. Cependant, nous établissons que pour certains ensembles partiellement ordonnés, l'équivalence a lieu en temps continu bien que n'étant pas vraie en temps discret.

We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.04.007
Paolo Dai Pra 1 ; Pierre-Yves Louis 2 ; Ida Minelli 1

1 Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy
2 Institut für Mathematik, Potsdam Universität, Am neuen Palais, 10 – Sans Souci, 14415 Potsdam, Germany 
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     title = {Monotonicity and complete monotonicity for continuous-time {Markov} chains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {965--970},
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     year = {2006},
     doi = {10.1016/j.crma.2006.04.007},
     language = {en},
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Paolo Dai Pra; Pierre-Yves Louis; Ida Minelli. Monotonicity and complete monotonicity for continuous-time Markov chains. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 965-970. doi : 10.1016/j.crma.2006.04.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.007/

[1] cdd/cdd+ software, 1995–2004, available at http://www.cs.mcgill.ca/~fukuda/soft/cdd_home/cdd.html

[2] P. Dai Pra, P.-Y. Louis, I. Minelli, Complete monotone coupling for Markov processes, in preparation

[3] J.A. Fill; M. Machida Stochastic monotonicity and realizable monotonicity, Ann. Probab., Volume 29 (2001) no. 2, pp. 938-978

[4] K. Fukuda; A. Prodon Double description method revisited, Combinatorics and Computer Science (Brest, 1995), Lecture Notes in Comput. Sci., vol. 1120, Springer, Berlin, 1996, pp. 91-111

[5] Gap software – groups, algorithms, programming – a system for computational discrete algebra, 1986–2006, available at http://www.gap-system.org/~gap/

[6] W.A. Massey Stochastic orderings for Markov processes on partially ordered spaces, Math. Oper. Res., Volume 12 (1987) no. 2, pp. 350-367

[7] J.G. Propp; D.B. Wilson Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms, Volume 9 (1996) no. 1–2, pp. 223-252

[8] Scilab software, 1989–2006. Copyright © INRIA ENPC, Scilab is a trademark of INRIA, available at http://www.scilab.org/

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