Advertisement

Levy Systems and Path Decompositions

  • J. W. Pitman
Chapter
Part of the Progress in Probability and Statistics book series (PRPR, volume 1)

Abstract

Itô [21] introduced the idea of a point process attached to a Markov process X, and subsequent work of Weil [42], Getoor [11], [12] and Maisonneuve [29] has shown that the existence of a suitably Markovian Lévy system for such a point process can be instrumental in establishing path decompositions of the Markov process. A path decomposition, or splitting time theorem, is a result to the effect that some fragment of the trajectory of X is conditionally independent of some other fragment given suitable conditioning variables, usually with one or more of the fragments being conditionally Markovian. Millar [32] gives a survey of such results, and more recent work may be found in the papers of Getoor, Pittenger, and Sharpe: [12], [14], [15], [16], [17], [18], [36], [37], [40]. Lévy systems suitable for deriving path decompositions were constructed in varying degrees of generality by Watanabe [41] and Benveniste and Jacod [2] for the point process of jumps, and by Itô [21], Dynkin [10] and Maisonneuve [28] for point processes of excursions.

Keywords

Markov Process Point Process Exit Time Predictable Process Path Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. BARLOW. Study of a filtration expanded to include an honest time. Z. Wahrscheinlichkeitstheorie verw. Gebiete 44 (1978), 307–323.CrossRefGoogle Scholar
  2. 2.
    A. BENVENISTE and J. JACOD. Systèmes de Lévy des processus de Markov. Invent. Math. 21 (1973), 183–198.CrossRefGoogle Scholar
  3. 3.
    P. BREMAUD. Point Processes and Queues: Martingale Dynamics. Forthcoming book.Google Scholar
  4. 4.
    K.L. CHUNG. On last exit times. Illinois J. Math. 4 (1960), 629–639.Google Scholar
  5. 5.
    K.L. CHUNG. On the boundary theory for Markov chains II. Acta. Math. 115 (1966), 111–163.CrossRefGoogle Scholar
  6. 6.
    K.L. CHUNG. Lectures on Boundary Theory for Markov Chains. Annals of Mathematics Studies Number 65, Princeton University Press, Princeton, 1970.Google Scholar
  7. 7.
    K.L. CHUNG. Excursions in Brownian motion. Arkiv. for Mat. 14 (1976), 155–177.CrossRefGoogle Scholar
  8. 8.
    P. COURREGE et P. PRIOURET. Temps d’arrêt d’une fonction aléatoire: relations d’equivalence associées et propriétés de décomposition. Publ. Inst. Statist. Univ. Paris 14 (1965), 245–274.Google Scholar
  9. 9.
    C. DELLACHERIE. Capacités et Processus Stochastiques. Springer-Verlag, Berlin, 1972.Google Scholar
  10. 10.
    E.B. DYNKIN. Wanderings of a Markov process. Theo. Prob. Appl. 16 (1971), 401–408.CrossRefGoogle Scholar
  11. 11.
    R.K. GETOOR. Excursions of a Markov process. Ann. Probab. 7 (1979), 244–266.CrossRefGoogle Scholar
  12. 12.
    R.K. GETOOR. Splitting times and shift functionals. Z. Wahrscheinlichkeitstheorie verw. Gebiete 47 (1979), 69–81.CrossRefGoogle Scholar
  13. 13.
    R.K. GETOOR and M.J. SHARPE. Last exit decompositions and distributions. Indiana Univ. Math. J. 23 (1973), 377–404.CrossRefGoogle Scholar
  14. 14.
    R.K. GETOOR and M.J. SHARPE. The Markov property at co-optional times. Z. Wahrscheinlichkeitstheorie verw. Gebiete 48 (1979), 201–211.CrossRefGoogle Scholar
  15. 15.
    R.K. GETOOR and M.J. SHARPE. Some random time dilations of a Markov process. Math. Zeit. 167 (1979), 187–199.CrossRefGoogle Scholar
  16. 16.
    R.K. GETOOR and M.J. SHARPE. Markov properties of a Markov process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55 (1981), 313–330.CrossRefGoogle Scholar
  17. 17.
    R.K. GETOOR and M.J. SHARPE. Excursions of dual processes. To appear.Google Scholar
  18. 18.
    R.K. GETOOR and M.J. SHARPE. Two results in dual excursions. In this volume.Google Scholar
  19. 19.
    P. GREENWOOD and J.W. PITMAN. Construction of local time and Poisson point processes from nested arrays. J. London Math. Soc. (2) 22 (1980), 182–192.CrossRefGoogle Scholar
  20. 20.
    P. GREENWOOD and J.W. PITMAN. Fluctuation identities for Levy processes and splitting at the maximum. Adv. Appl. Prob. 12 (1980), 893–902.CrossRefGoogle Scholar
  21. 21.
    K. ITO. Poisson point processes attached to Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. pp. 225–239. Univ. of California Press, Berkeley, 1970.Google Scholar
  22. 22.
    J. JACOD. Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714, Springer-Verlag, Berlin, 1979.Google Scholar
  23. 23.
    T. JEULIN. Semi-Martingales et Grossissement d’une Filtration. Lecture Notes in Math. 833, Springer-Verlag, Berlin, 1980.Google Scholar
  24. 24.
    T. JEULIN et M. YOR. Nouveaux résultats sur le grossissement des tribus. Ann. Sci. E.N.S. 4 e Série, t. 11 (1978), 429–443.Google Scholar
  25. 25.
    T. JEULIN et M. YOR. Sur les distributions de certaines fonctionelles du mouvement brownien. Séminaire de Probabilités XV (Univ. Strasbourg), pp. 210–226. Lecture Notes Math. 850 Springer-Verlag, Berlin, 1981.Google Scholar
  26. 26.
    B. MAISONNEUVE. Ensembles régénératifs, temps locaux et subordinateurs. Séminaire de Probabilités V (Univ. Strasbourg), pp. 147–169. Lecture Notes Math. 191. Springer-Verlag, Berlin, 1971.Google Scholar
  27. 27.
    B. MAISONNEUVE. Systèmes régénératifs. Astérisque, No. 15, Soc. Math. France, Paris, 1974.Google Scholar
  28. 28.
    B. MAISONNEUVE. Exit systems. Ann. Prob. 3 (1975), 399–411.CrossRefGoogle Scholar
  29. 29.
    B. MAISONNEUVE. On the structure of certain excursions of a Markov process. Z. Wahrshceinlichkeitstheorie verw. Gebiete 47 (1979), 61–67.CrossRefGoogle Scholar
  30. 30.
    P.A. MEYER. Processus de Poisson ponctuels, d’aprés K. Itô. Séminaire de Probabilités V (Univ. Strasbourg), pp. 177–190. Lecture Notes Math. 191, Springer-Verlag, Berlin, 1971.Google Scholar
  31. 31.
    P.A. MEYER, R.T. SMYTHE, and J.B. WALSH. Birth and death of Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. vol. 3, pp. 295–306. Univ. California Press, Berkeley, 1972.Google Scholar
  32. 32.
    P.W. MILLAR. Random times and decomposition theorems. Proc. Symp. Pure Math. vol. 31, Providence, 1976.Google Scholar
  33. 33.
    P.W. MILLAR. A path decomposition for Markov processes. Ann. Prob. 6 (1978), 345–348.CrossRefGoogle Scholar
  34. 34.
    J.W. PITMAN. Occupation measures for Markov chains. Adv. Appl. Prob. 9 (1977), 69–86.CrossRefGoogle Scholar
  35. 35.
    J.W. PITMAN and M. YOR. A decomposition of Bessel bridges. To appear.Google Scholar
  36. 36.
    A.O. PITTENGER. Regular birth times for Markov processes. Ann. Prob. To appear.Google Scholar
  37. 37.
    A.O. PITTENGER and M.J. SHARPE. Regular birth and death times. Z. Wahrscheinlichkeitstheorie verw. Gebiete, to appear.Google Scholar
  38. 38.
    A.O. PITTENGER and C.T. SHIH. Coterminal familes and the strong Markov property. Trans. Amer. Math. Soc. 182 (1973), 1–42.CrossRefGoogle Scholar
  39. 39.
    L.C.G. ROGERS. Williams’ characterization of the Brownian excursion law: proof and applications. Séminaire de Probabilités XV (Univ. Strasbourg), pp. 227–250. Lecture Notes Math. 850. Springer-Verlag, Berlin, 1981.Google Scholar
  40. 40.
    M.J. SHARPE. Killing times for Markov processes. To appear.Google Scholar
  41. 41.
    S. WATANABE. On discontinuous additive functionals and Lévy measures of a Markov process. Japan J. Math. 34 (1964), 53–79.Google Scholar
  42. 42.
    M. WEIL. Conditionnement par rapport au passé strict. Séminaire de Probabilités V (Univ. Strasbourg), pp. 362–372. Lecture Notes Math. 191. Springer-Verlag, Berlin, 1971.Google Scholar
  43. 42.
    D. WILLIAMS. Path decomposition and continuity of local time for one-dimensional diffusions. Proc. London Math. Soc. 28 (1974), 738–768.CrossRefGoogle Scholar
  44. 43.
    D. WILLIAMS. Diffusions, Markov Processes, and Martingales; vol. 1: Foundations. Wiley, New York, 1979.Google Scholar

Copyright information

© Birkhäuser Boston 1981

Authors and Affiliations

  • J. W. Pitman
    • 1
  1. 1.Department of StatisticsUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations

Cite chapter

Buy options