Probability Theory
Monotonicity and complete monotonicity for continuous-time Markov chains
Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 965-970.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.04.007
Dai Pra, Paolo 1; Louis, Pierre-Yves 2; Minelli, Ida 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
@article{CRMATH_2006__342_12_965_0,
     author = {Dai Pra, Paolo and Louis, Pierre-Yves and Minelli, Ida},
     title = {Monotonicity and complete monotonicity for continuous-time {Markov} chains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {965--970},
     publisher = {Elsevier},
     volume = {342},
     number = {12},
     year = {2006},
     doi = {10.1016/j.crma.2006.04.007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.04.007/}
}
TY  - JOUR
AU  - Dai Pra, Paolo
AU  - Louis, Pierre-Yves
AU  - Minelli, Ida
TI  - Monotonicity and complete monotonicity for continuous-time Markov chains
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 965
EP  - 970
VL  - 342
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.04.007/
DO  - 10.1016/j.crma.2006.04.007
LA  - en
ID  - CRMATH_2006__342_12_965_0
ER  - 
%0 Journal Article
%A Dai Pra, Paolo
%A Louis, Pierre-Yves
%A Minelli, Ida
%T Monotonicity and complete monotonicity for continuous-time Markov chains
%J Comptes Rendus. Mathématique
%D 2006
%P 965-970
%V 342
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.04.007/
%R 10.1016/j.crma.2006.04.007
%G en
%F CRMATH_2006__342_12_965_0
Dai Pra, Paolo; Louis, Pierre-Yves; Minelli, Ida. 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. http://www.numdam.org/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] Fill, J.A.; Machida, M. Stochastic monotonicity and realizable monotonicity, Ann. Probab., Volume 29 (2001) no. 2, pp. 938-978

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

[7] Propp, J.G.; Wilson, D.B. 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/

Cited by Sources: