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 Electronic Communications in Probability > Vol. 11 (2006) > Paper 25 open journal systems 


On the Existence of Recurrent Extensions of Self-similar Markov Processes

Patrick J Fitzsimmons, UC San Diego


Abstract
Let X = (Xt)t ≥ 0 be a self-similar Markov process with values in the non-negative half-line, such that the state 0 is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. This condition is expressed in terms of the Lévy process associated with X by the Lamperti transformation.


Full text: PDF

Pages: 230-241

Published on: October 11, 2006
Bibliography
  1. J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005), 191-212. Math. Review MR2178044
  2. R. M. Blumenthal. On construction of Markov processes. Z. Wahrsch. Verw. Gebiete 63 (1983), 433-444. Math. Review MR705615
  3. R. M. Blumenthal. Excursions of Markov Processes. Birkhäuser, Boston, 1992. Math. Review MR1138461
  4. L. Chaumont and V. Rivero. On some transformations between positive self-similar Markov processes. Preprint, 2006. Math. Review number not available.
  5. E. B. Dynkin. An application of flows to time shift and time reversal in stochastic processes. Trans. Amer. Math. Soc. 287 (1985), 613-619. Math. Review MR768728
  6. P. J. Fitzsimmons. Homogeneous random measures and a weak order for the excessive measures of a Markov process. Trans. Amer. Math. Soc. 303 (1987), 431-478. Math. Review MR902778
  7. P. J. Fitzsimmons. On the excursions of Markov processes in classical duality. Probab. Theory Related Fields 75 (1987), 159-178. Math. Review MR885460
  8. P. J. Fitzsimmons. Markov processes with equal capacities. J. Theoret. Probab. 12 (1999), 271-292. Math. Review MR1675005
  9. P. J. Fitzsimmons and R. K. Getoor. Revuz measures and time changes. Math. Z. 199 (1988), 233-256. Math. Review MR958650
  10. P.J. Fitzsimmons and R.K. Getoor. Excursion theory revisited. Illinois J. Math. 50 (2006), 413-437. Math. Review number not available.
  11. R. K. Getoor. Excessive measures. Birkhäuser, Boston, 1990. Math. Review MR1093669
  12. R. K. Getoor and J. Glover. Constructing Markov processes with random times of birth and death. In Seminar on stochastic processes, 1986, pages 35-69. Birkhäuser, Boston, 1987. Math. Review MR902426
  13. K. Itô. Poisson point processes attached to Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, (Vol. III), pages 225-239. Univ. California Press, Berkeley, 1972. Math. Review MR0402949
  14. H. Kaspi. Random time changes for processes with random birth and death. Ann. Probab. 16 (1988), 586-599. Math. Review MR929064
  15. J. Lamperti. Semi-stable Markov processes, I. Z. Wahrsch. verw. Gebiete 22 (1972), 205-225. Math. Review MR0307358
  16. B. Maisonneuve. Exit systems. Ann. Probability 3 (1975), 399-411. Math. Review MR0400417
  17. J. B. Mitro. Dual Markov processes: construction of a useful auxiliary process. Z. Wahrsch. Verw. Gebiete 47 (1979), 139-156. Math. Review MR523166
  18. V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11 (2005), 471-509. Math. Review MR2146891
  19. V. Rivero. On recurrent extensions of positive self similar Markov processes and Cramér's condition II. Preprint, 2006. Math. Review number not available.
  20. T. S. Salisbury. On the Itô excursion process. Probab. Theory Related Fields 73 (1986), 319-350. Math. Review MR859837
  21. T. S. Salisbury. Construction of right processes from excursions. Probab. Theory Related Fields 73 (1986), 351-367. Math. Review MR859838
  22. J. Vuolle-Apiala. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994), 546-565. Math. Review MR1288123
  23. J. Vuolle-Apiala and S. E. Graversen. Duality theory for self-similar processes. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), 323-332. Math. Review MR871085
















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Electronic Communications in Probability. ISSN: 1083-589X