Regular Birth and Death Times for Markov Processes

  • A. O. Pittenger
Part of the Progress in Probability and Statistics book series (PRPR, volume 1)


The concept underlying a homogeneous Markov process is that the evolution of the process from time t onwards depends only on the position at time t and not on the past before t. There are two aspects to this idea. First, there is the conditional independence of the t-past and t-future given the t-present, and second, there is the subsequent evolution of the process as a homogeneous Markov process obeying the same laws of transition. Both aspects are meant by the assertion that the process is Markov or has the Markov property.


Markov Process Conditional Independence Random Time Markov Property Terminal Time 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. AZEMA. Théorie générale des processus et retournement du temps. Ann. Sci. Ecole Norm. Sup. 4 Série t6 (1973), 459–519.Google Scholar
  2. 2.
    R.M. BLUMENTHAL. An extended Markov property. Trans. Amer. Math. Soc. 85 (1957), 52–72.CrossRefGoogle Scholar
  3. 3.
    R.M. BLUMENTHAL and R.K. Getoor. Markov Processes and Potential Theory. Academic Press, New York, 1968.Google Scholar
  4. 4.
    J.L. DOOB. Markoff chains — denumerable case. Trans. Amer. Math. Soc. 58 (1945), 455–473.Google Scholar
  5. 5.
    E.B. DYNKIN and A.A. YUSHKEVICH. Strong Markov processes. Theory Prob. and Appl. 1 (1956), 134–139.CrossRefGoogle Scholar
  6. 6.
    R.K. GETOOR. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math. 440. Springer, Berlin 1975.Google Scholar
  7. 7.
    R.K. GETOOR and M.J. SHARPE. The Markov property at co-optional times. Z. Wahrscheinlichkeitstheorie verw. Gebiete 48 (1979), 201–211.CrossRefGoogle Scholar
  8. 8.
    R.K. GETOOR and M.J. SHARPE. Markov properties of a Markov process. Z. Wahrscheinlichkeitstheorie verw Gebiete 55 (1981), 313–330.CrossRefGoogle Scholar
  9. 9.
    G.A. HUNT. Some theorems concerning Brownian motion. Trans. Amer. Math. Soc. 81 (1956), 294–319.CrossRefGoogle Scholar
  10. 10.
    M. JACOBSEN. Markov chains: birth and death times with conditional independence. Preprint.Google Scholar
  11. 11.
    M. JACOBSEN and J. PITMAN. Birth, death, and conditioning of Markov chains. Ann. Prob. 5 (1977), 430–450.CrossRefGoogle Scholar
  12. 12.
    B. MAISONNEUVE. Exit systems. Ann. Prob. 3 (1975), 399–411.CrossRefGoogle Scholar
  13. 13.
    P.A. MEYER, R. SMYTHE, J.B. WALSH. Birth and death of a Markov process. Proc. Sixth Berkeley Symp. Math. Stat. Prob., Vol. III. 295–305. University of California Press, Berkeley, 1972.Google Scholar
  14. 14.
    P.W. MILLAR. Random times and decomposition theorems. AMS Proc. of Symposia in Pure Math. 31 (1977), 91–103.Google Scholar
  15. 15.
    M. NAGASAWA. Time reversion of Markov processes. Nagoya Math. J. 24 (1964), 177–204.Google Scholar
  16. 16.
    A.O. PITTENGER. Regular birth times for Markov processes. Ann. Prob., to appear.Google Scholar
  17. 17.
    A.O. PITTENGER and M.J. Sharpe. Regular birth and death times. Z. Wahrscheinlichkeitstheorie verw. Gebiete, to appear.Google Scholar
  18. 18.
    A.O. PITTENGER and C.T. SHIH. Coterminal families and the strong Markov property. Trans. Amer. Math. Soc. 182 (1973), 1–42.CrossRefGoogle Scholar
  19. 19.
    M.J. SHARPE. Killing times for Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, to appear.Google Scholar
  20. 20.
    M.J. SHARPE. General theory of Markov processes. Forthcoming book.Google Scholar
  21. 21.
    J.B. WALSH. The perfection of multiplicative functionals. Séminaire des Probabilités VI (Univ. de Strasbourg), pp. 233–242. Lecture Notes in Math. 258. Springer-Verlag, Berlin, 1972.Google Scholar

Copyright information

© Birkhäuser Boston 1981

Authors and Affiliations

  • A. O. Pittenger
    • 1
  1. 1.Department of MathematicsUniversity of Maryland, Baltimore CountyCatonsvilleUSA

Personalised recommendations