Lévy processes conditioned on having a large height process

Mathieu Richard

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 982-1013
  • ISSN: 0246-0203

Abstract

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In the present work, we consider spectrally positive Lévy processes ( X t , t ≥ 0 ) not drifting to + ∞ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with X ) before hitting 0 . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ℙ x ☆ of this conditioned process (starting at x g t ; 0 ) is defined as a Doob h -transform via a martingale. For Lévy processes with infinite variation paths, this martingale is ( ∫ ρ ˜ t ( d z ) e α z + I t ) 1 { t ≤ T 0 } for some α and where ( I t , t ≥ 0 ) is the past infimum process of X , where ( ρ ˜ t , t ≥ 0 ) is the so-calledexploration processdefined in [10] and where T 0 is the hitting time of 0 for X . Under ℙ x ☆ , we also obtain a path decomposition of X at its minimum, which enables us to prove the convergence of ℙ x ☆ as x → 0 . When the process X is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of X . The computations are easier in this case because X can be viewed as the contour process of a (sub)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

How to cite

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Richard, Mathieu. "Lévy processes conditioned on having a large height process." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 982-1013. <http://eudml.org/doc/271942>.

@article{Richard2013,
abstract = {In the present work, we consider spectrally positive Lévy processes $(X_\{t\},t\ge 0)$ not drifting to $+\infty $ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $\{\mathbb \{P\}\}_\{x\}^\{\star \}$ of this conditioned process (starting at $x&gt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int \tilde\{\rho \}_\{t\}(\mathrm \{d\}z)\mathrm \{e\}^\{\{\alpha \}\{z\}\}+I_\{t\})\mathbf \{1\} _\{\lbrace t\le T_\{0\}\rbrace \}$ for some $\alpha $ and where $(I_\{t\},t\ge 0)$ is the past infimum process of $X$, where $(\tilde\{\rho \}_\{t\},t\ge 0)$ is the so-calledexploration processdefined in [10] and where $T_\{0\}$ is the hitting time of 0 for $X$. Under $\{\mathbb \{P\}\}_\{x\}^\{\star \}$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\{\mathbb \{P\}\}_\{x\}^\{\star \}$ as $x\rightarrow 0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.},
author = {Richard, Mathieu},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy process; height process; Doob harmonic transform; splitting tree; spine decomposition; Size-biased distribution; queueing theory; size-biased distribution; exploration process},
language = {eng},
number = {4},
pages = {982-1013},
publisher = {Gauthier-Villars},
title = {Lévy processes conditioned on having a large height process},
url = {http://eudml.org/doc/271942},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Richard, Mathieu
TI - Lévy processes conditioned on having a large height process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 982
EP - 1013
AB - In the present work, we consider spectrally positive Lévy processes $(X_{t},t\ge 0)$ not drifting to $+\infty $ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb {P}}_{x}^{\star }$ of this conditioned process (starting at $x&gt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int \tilde{\rho }_{t}(\mathrm {d}z)\mathrm {e}^{{\alpha }{z}}+I_{t})\mathbf {1} _{\lbrace t\le T_{0}\rbrace }$ for some $\alpha $ and where $(I_{t},t\ge 0)$ is the past infimum process of $X$, where $(\tilde{\rho }_{t},t\ge 0)$ is the so-calledexploration processdefined in [10] and where $T_{0}$ is the hitting time of 0 for $X$. Under ${\mathbb {P}}_{x}^{\star }$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of ${\mathbb {P}}_{x}^{\star }$ as $x\rightarrow 0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)criticalsplitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.
LA - eng
KW - Lévy process; height process; Doob harmonic transform; splitting tree; spine decomposition; Size-biased distribution; queueing theory; size-biased distribution; exploration process
UR - http://eudml.org/doc/271942
ER -

References

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  1. [1] R. Abraham and J.-F. Delmas. Feller property and infinitesimal generator of the exploration process. J. Theoret. Probab.20 (2007) 355–370. Zbl1129.60075MR2324536
  2. [2] D. Aldous. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab.1 (1991) 228–266. Zbl0733.60016MR1102319
  3. [3] K. B. Athreya and P. E. Ney. Branching Processes. Grundlehren der mathematischen Wissenschaften 196. Springer, New York, 1972. MR373040
  4. [4] J. Azéma and M. Yor. Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) 90–115. Lecture Notes in Math. 721. Springer, Berlin, 1979. MR544782
  5. [5] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996. MR1406564
  6. [6] L. Chaumont. Sur certains processus de Lévy conditionnés à rester positifs. Stochastics Stochastics Rep.47 (1994) 1–20. MR1787140
  7. [7] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl.64 (1996) 39–54. Zbl0879.60072MR1419491
  8. [8] L. Chaumont and R. A. Doney. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005) 948–961 (electronic). Zbl1109.60039MR2164035
  9. [9] T. Duquesne. Continuum random trees and branching processes with immigration. Stochastic Process. Appl.119 (2009) 99–129. Zbl1161.60028MR2485021
  10. [10] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. Zbl1037.60074MR1954248
  11. [11] J. Geiger. Size-biased and conditioned random splitting trees. Stochastic Process. Appl.65 (1996) 187–207. Zbl0889.60089MR1425355
  12. [12] J. Geiger. Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Probab.36 (1999) 301–309. Zbl0942.60071MR1724856
  13. [13] J. Geiger and G. Kersting. Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 111–126. IMA Vol. Math. Appl. 84. Springer, New York, 1997. Zbl0867.60061MR1601713
  14. [14] K. Hirano. Lévy processes with negative drift conditioned to stay positive. Tokyo J. Math.24 (2001) 291–308. Zbl1020.60040MR1844435
  15. [15] D. P. Kennedy. Some martingales related to cumulative sum tests and single-server queues. Stochastic Processes Appl.4 (1976) 261–269. Zbl0338.60030MR420834
  16. [16] A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields122 (2002) 42–70. Zbl1020.60074MR1883717
  17. [17] A. Lambert. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab.12 (2007) 420–446. Zbl1127.60082MR2299923
  18. [18] A. Lambert. Population dynamics and random genealogies. Stoch. Models24 (2008) 45–163. Zbl05383029MR2466449
  19. [19] A. Lambert. The contour of splitting trees is a Lévy process. Ann. Probab.38 (2010) 348–395. Zbl1190.60083MR2599603
  20. [20] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab.26 (1998) 213–252. Zbl0948.60071MR1617047
  21. [21] Z.-H. Li. Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A68 (2000) 68–84. Zbl0960.60072MR1727226
  22. [22] V. Limic. A LIFO queue in heavy traffic. Ann. Appl. Probab.11 (2001) 301–331. Zbl1015.60079MR1843048
  23. [23] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab.23 (1995) 1125–1138. Zbl0840.60077MR1349164
  24. [24] P. Millar. Zero–one laws and the minimum of a Markov process. Trans. Am. Math. Soc.226 (1977) 365–391. Zbl0381.60062MR433606
  25. [25] L. Nguyen-Ngoc. Limiting laws and penalization of certain Lévy processes by a function of their maximum. Teor. Veroyatn. Primen.55 (2010) 530–547. MR2768536
  26. [26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der mathematischen Wissenschaften 293. Springer, Berlin, 1999. Zbl0917.60006MR1725357
  27. [27] P. Robert. Stochastic Networks and Queues, french edition. Applications of Mathematics (New York) 52. Stochastic Modelling and Applied Probability. Springer, Berlin, 2003. MR1996883
  28. [28] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795–798. MR22045

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