(Homogeneous) markovian bridges

Vincent Vigon

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 3, page 875-916
  • ISSN: 0246-0203

Abstract

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(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).

How to cite

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Vigon, Vincent. "(Homogeneous) markovian bridges." Annales de l'I.H.P. Probabilités et statistiques 47.3 (2011): 875-916. <http://eudml.org/doc/242908>.

@article{Vigon2011,
abstract = {(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).},
author = {Vigon, Vincent},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Markov chains; random walks; LU-factorization; path-decomposition; fluctuation theory; probabilistic potential theory; infinite matrices; Martin boundary; Markov bridges; Wiener-Hopf factorization},
language = {eng},
number = {3},
pages = {875-916},
publisher = {Gauthier-Villars},
title = {(Homogeneous) markovian bridges},
url = {http://eudml.org/doc/242908},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Vigon, Vincent
TI - (Homogeneous) markovian bridges
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 3
SP - 875
EP - 916
AB - (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).
LA - eng
KW - Markov chains; random walks; LU-factorization; path-decomposition; fluctuation theory; probabilistic potential theory; infinite matrices; Martin boundary; Markov bridges; Wiener-Hopf factorization
UR - http://eudml.org/doc/242908
ER -

References

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  1. [1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1998. Zbl0938.60005MR1406564
  2. [2] J. D. Biggins. Random walk conditioned to stay positive. J. London Math. Soc. (2) 67 (2003) 259–272. Zbl1046.60066MR1942425
  3. [3] R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York–London, 1968. Zbl0169.49204MR264757
  4. [4] C. Dellacherie and P.-A. Meyer. 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. Zbl0526.60001MR727641
  5. [5] J. L. Doob. Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85 (1957) 431–458. Zbl0097.34004MR109961
  6. [6] E. B. Dynkin. Boundary theory of Markov processes (the discrete case). Russian Math. Surveys 24 (1969) 1–42. Zbl0222.60048MR245096
  7. [7] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, 1966. Zbl0155.23101MR210154
  8. [8] P. J. Fitzsimmons. On the excursions of Markov processes in classical duality. Probab. Theory Related Fields 75 (1987) 159–178. Zbl0616.60070MR885460
  9. [9] P. Fitzsimmons. Markov processes with identical bridges. Electron. J. Probab. 3 (1998). Zbl0907.60066MR1641066
  10. [10] P. Fitzsimmons, J. Pitman and M. Yor. Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes 101–134. Progress in Probability 33. Birkhäuser Boston, Boston, 1992. Zbl0844.60054MR1278079
  11. [11] S. Fourati. Vervaat et Lévy. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 461–478. Zbl1074.60082MR2139029
  12. [12] R. K. Getoor and M. J. Sharpe. The Markov property at co-optional times. Z. Wahrsch. Verw. Gebiete 48 (1979) 201–211. Zbl0402.60066MR534845
  13. [13] R. K. Getoor and M. J. Sharpe. Excursions of dual processes. Adv. Math. 45 (1982) 259–309. Zbl0497.60067MR673804
  14. [14] J. Hoffmann-Jørgensen. Markov sets. Math. Scand. 24 (1969) 145–166. Zbl0232.60053MR256460
  15. [15] G. A. Hunt. Markoff processes and potentials. Illinois J. Math. 2 (1958) 151–213. Zbl0100.13804MR107097
  16. [16] G. A. Hunt. Markoff chains and Martin boundaries. Illinois J. Math. 4 (1960) 316–340. Zbl0094.32103MR123364
  17. [17] K. Itô. 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. Zbl0284.60051MR402949
  18. [18] M. Jacobsen. Splitting times for Markov processes and a generalised Markov property for diffusions. Z. Wahrsch. Verw. Gebiete 30 (1974) 27–43. Zbl0288.60064MR375477
  19. [19] M. Jacobsen and J. M. Pitman. Birth, death and conditioning of Markov chains. Ann. Probab. 5 (1977) 430–450. Zbl0363.60052MR445613
  20. [20] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer, New York, 2002. Zbl0892.60001MR1876169
  21. [21] A. N. Kolmogorov. Zur Theorie der Markoffschen Ketten. Math. Ann. 112 (1936) 155–160. Zbl0012.41001MR1513044
  22. [22] B. Maisonneuve. Sytèmes régénératifs. Astérique 15. Société mathématique de France, 1974. Zbl0285.60049MR350879
  23. [23] P. A. Meyer, R. T. Smythe and J. B. Walsh. 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. Zbl0255.60046MR405600
  24. [24] P. W. Millar. Zero–one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 (1977) 365–391. Zbl0381.60062MR433606
  25. [25] M. Nagasawa. Time reversions of Markov processes. Nagoya Math. J. 24 (1964) 177–204. Zbl0133.10702MR169290
  26. [26] J. Pitman and M. Yor. Itô’s excursion theory and its applications. Japan J. Math. 2 (2007) 83–96. Zbl1156.60066MR2295611
  27. [27] A. O. Pittenger and C. T. Shih. Coterminal families and the strong Markov property. Trans. Amer. Math. Soc. 182 (1973) 1–42. Zbl0275.60084MR336827
  28. [28] H. Rost. Markoff–Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier (Grenoble) 21 (1971) 253–270. Zbl0197.44206MR299755
  29. [29] E. Seneta. Non-Negative Matrices. An Introduction to Theory and Applications. Allen & Unwin, London, 1973. Zbl0278.15011MR389944
  30. [30] T. Simon. Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields 115 (1999) 445–477. Zbl0944.60049MR1728917
  31. [31] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and Its Applications (New York). Springer, New York, 2000. Zbl1044.60510MR1741181
  32. [32] W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 141–149. Zbl0392.60058MR515820
  33. [33] V. Vigon. Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf. Editions Universitaires Europeennes, also disposable on HAL and on my web page, 2002. 
  34. [34] D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 (1970) 871–873. Zbl0233.60066MR258130
  35. [35] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge, 2000. Zbl0951.60002MR1743100

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