Les ponts markoviens (homogènes) sont des chaines de Markov (homogènes) qui démarrent à un point donné et meurent à un point donné. Pour préserver l'homogénéité, une telle chaine de Markov a nécessairement une durée de vie aléatoire. Nous étudions les ponts pour eux mêmes et pour leur utilité à décrire les transformations d'une chaine de Markov : restriction à un intervalle aléatoire, renversement temporel, changement de temps, conditionnements variés : notamment le confinement dans une partie de l'espace d'état. Ces ponts nous conduisent à considérer les chaines de Markov d'un point de vue inhabituel : nous ne travaillons plus avec une seule matrice de transition comme à l'accoutumée, mais avec une classe de matrices qui se déduisent les unes des autres par transformation de Doob. Cette méthode a l'avantage de mieux décrire les symétries passé ↔ futur : symétrie de l'indépendance conditionnelle (bien connue) et symétrie de l'homogénéité (moins bien connue).
(Homogeneous) Markov bridges are (time homogeneous) Markov chains which begin at a given point and end at a given point. The price to pay for preserving the homogeneity is to work with processes with a random life-span. Bridges are studied both for themselves and for their use in describing the transformations of Markov chains: restriction on a random interval, time reversal, time change, various conditionings comprising the confinement in some part of the state space. These bridges lead us to look at Markov chains from an unusual point of view: we will work, no longer with only one transition matrix, but with a class of matrices which can be deduced one from the other by Doob transformations. This way of proceeding has the advantage of better describing the “past ↔ future symmetries”: The symmetry of conditional independence (well known) and the symmetry of homogeneity (less well known).
@article{AIHPB_2011__47_3_875_0,
author = {Vigon, Vincent},
title = {(Homogeneous) markovian bridges},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
volume = {47},
year = {2011},
pages = {875-916},
doi = {10.1214/10-AIHP391},
mrnumber = {2841078},
zbl = {1267.60080},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPB_2011__47_3_875_0}
}
Vigon, Vincent. (Homogeneous) markovian bridges. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) pp. 875-916. doi : 10.1214/10-AIHP391. http://gdmltest.u-ga.fr/item/AIHPB_2011__47_3_875_0/
[1] . Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1998. | MR 1406564 | Zbl 0938.60005
[2] . Random walk conditioned to stay positive. J. London Math. Soc. (2) 67 (2003) 259-272. | MR 1942425 | Zbl 1046.60066
[3] and . Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York-London, 1968. | MR 264757 | Zbl 0169.49204
[4] and . Probabilités et potentiel. Chapitres IX à XI: Théorie discrète du potentiel. Publications de l'Institut de Mathématique de l'Université de Strasbourg XVIII. Actualités Scientifiques et Industrielles 1410. Hermann, Paris, 1983. | MR 727641 | Zbl 0526.60001
[5] . Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85 (1957) 431-458. | Numdam | MR 109961 | Zbl 0097.34004
[6] . Boundary theory of Markov processes (the discrete case). Russian Math. Surveys 24 (1969) 1-42. | MR 245096 | Zbl 0222.60048
[7] . An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, 1966. | MR 210154 | Zbl 0155.23101
[8] . On the excursions of Markov processes in classical duality. Probab. Theory Related Fields 75 (1987) 159-178. | MR 885460 | Zbl 0616.60070
[9] . Markov processes with identical bridges. Electron. J. Probab. 3 (1998). | MR 1641066 | Zbl 0907.60066
[10] , and . Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes 101-134. Progress in Probability 33. Birkhäuser Boston, Boston, 1992. | MR 1278079 | Zbl 0844.60054
[11] . Vervaat et Lévy. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 461-478. | Numdam | MR 2139029 | Zbl 1074.60082
[12] and . The Markov property at co-optional times. Z. Wahrsch. Verw. Gebiete 48 (1979) 201-211. | MR 534845 | Zbl 0402.60066
[13] and . Excursions of dual processes. Adv. Math. 45 (1982) 259-309. | MR 673804 | Zbl 0497.60067
[14] . Markov sets. Math. Scand. 24 (1969) 145-166. | MR 256460 | Zbl 0232.60053
[15] . Markoff processes and potentials. Illinois J. Math. 2 (1958) 151-213. | MR 107097 | Zbl 0100.13804
[16] . Markoff chains and Martin boundaries. Illinois J. Math. 4 (1960) 316-340. | MR 123364 | Zbl 0094.32103
[17] . Poisson point processes attached to Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 225-239. Univ. California Press, Berkeley, 1972. | MR 402949 | Zbl 0284.60051
[18] . Splitting times for Markov processes and a generalised Markov property for diffusions. Z. Wahrsch. Verw. Gebiete 30 (1974) 27-43. | MR 375477 | Zbl 0288.60064
[19] and . Birth, death and conditioning of Markov chains. Ann. Probab. 5 (1977) 430-450. | MR 445613 | Zbl 0363.60052
[20] . Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer, New York, 2002. | MR 1876169 | Zbl 0892.60001
[21] . Zur Theorie der Markoffschen Ketten. Math. Ann. 112 (1936) 155-160. | MR 1513044 | Zbl 0012.41001
[22] . Sytèmes régénératifs. Astérique 15. Société mathématique de France, 1974. | MR 350879 | Zbl 0285.60049
[23] , and . Birth and death of Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 295-305. Univ. California Press, Berkeley, 1972. | MR 405600 | Zbl 0255.60046
[24] . Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 (1977) 365-391. | MR 433606 | Zbl 0381.60062
[25] . Time reversions of Markov processes. Nagoya Math. J. 24 (1964) 177-204. | MR 169290 | Zbl 0133.10702
[26] and . Itô's excursion theory and its applications. Japan J. Math. 2 (2007) 83-96. | MR 2295611 | Zbl 1156.60066
[27] and . Coterminal families and the strong Markov property. Trans. Amer. Math. Soc. 182 (1973) 1-42. | MR 336827 | Zbl 0275.60084
[28] . Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier (Grenoble) 21 (1971) 253-270. | Numdam | MR 299755 | Zbl 0197.44206
[29] . Non-Negative Matrices. An Introduction to Theory and Applications. Allen & Unwin, London, 1973. | MR 389944 | Zbl 0278.15011
[30] . Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields 115 (1999) 445-477. | MR 1728917 | Zbl 0944.60049
[31] . Coupling, Stationarity, and Regeneration. Probability and Its Applications (New York). Springer, New York, 2000. | MR 1741181 | Zbl 1044.60510
[32] . A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 141-149. | MR 515820 | Zbl 0392.60058
[33] . Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Editions Universitaires Europeennes, also disposable on HAL and on my web page, 2002.
[34] . Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 (1970) 871-873. | MR 258130 | Zbl 0233.60066
[35] . Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge, 2000. | MR 1743100 | Zbl 0951.60002