Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 10 (2005) > Paper 28 open journal systems 


On Lévy processes conditioned to stay positive.

Loïc Chaumont, LPMA - Université Paris 6
Ronald Arthur Doney, Department of Mathematics- University of Manchester


Abstract
We construct the law of Lévy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of Lévy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying Lévy process reflected at its minimum. Then, when the Lévy process creeps upwards, we study the lower tail at 0 of the law of the height of this excursion.


Full text: PDF

Pages: 948-961

Published on: July 14, 2005
Bibliography
  1. J. Bertoin: Lévy Processes, Cambridge University Press, Cambridge, (1996).Math. Review 98e:60117
  2. J. Bertoin: An extension of Pitman's theorem for spectrally positive Lévy processes. Ann. Probab. 20, no. 3, 1464--1483, (1992).Math. Review 93k:60191
  3. J. Bertoin: Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stoch. Process, Appl. 47, 17-35, (1993).Math. Review 94h:60110
  4. J. Bertoin and R.A. Doney: On conditioning a random walk to stay nonnegative. Ann. Probab. 22, no. 4, 2152-2167, (1994).Math. Review 96b:60168
  5. L. Chaumont: Sur certains processus de Lévy conditionnés à rester positifs. Stochastics and Stoch. Rep., 47, 1--20, (1994).Math. Review 2001e:60093
  6. L. Chaumont: Conditionings and path decompositions for Lévy processes. Stoch. Process. Appl. 64, 39-54, (1996).Math. Review 98b:60131
  7. R.A Doney: Tanaka's construction of Lévy processes conditioned to stay positive. Séminaire de Probabilités, XXXVIII, 1--4, Lecture Notes in Math., 1686, Springer, Berlin, 2005. Math. Review number not available.
  8. R.A Doney: Some excursion calculations for spectrally one-sided Lévy processes. Séminaire de Probabilités, XXXVIII, 5--15, Lecture Notes in Math., 1686, Springer, Berlin, 2005. Math. Review number not available.
  9. T. Duquesne: Path decompositions for real Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 39, no. 2, 339--370, (2003).Math. Review 2004b:60125
  10. K. Hirano: Lévy processes with negative drift conditioned to stay positive. Tokyo J. Math., 24, 291-308, (2001).Math. Review 2003e:60099
  11. P.W. Millar: Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226, 365--391, (1977).Math. Review 55 #6579
  12. V. Rivero: Recouvrements aléatoires et processus de Markov auto-similaires. PhD Thesis. Université Paris 6, (2004). Math. Review number not available.
  13. M.L. Silverstein: Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8, 539-575, (1980).Math. Review 81f:60058
  14. H. Tanaka: Time reversal of random walks in one dimension. Tokyo J. Math. 12, 159-174, (1989).Math. Review 90m:60083
  15. H. Tanaka: Lévy processes conditioned to stay positive and diffusions in random environments. Advanced Studies in Pure Mathematics, 39, 355--376, (2004).Math. Review 2005e:60240
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489