A remarkable σ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano

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

  • Volume: 49, Issue: 4, page 1014-1032
  • ISSN: 0246-0203

Abstract

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The σ -finite measure 𝒫 sup which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s h -transform processes with respect to these functions are utilized for the construction of 𝒫 sup .

How to cite

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Yano, Yuko. "A remarkable $\sigma $-finite measure unifying supremum penalisations for a stable Lévy process." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1014-1032. <http://eudml.org/doc/272042>.

@article{Yano2013,
abstract = {The $\sigma $-finite measure $\mathcal \{P\} _\{\sup \}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$-transform processes with respect to these functions are utilized for the construction of $\mathcal \{P\} _\{\sup \}$.},
author = {Yano, Yuko},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy processes; stable Lévy processes; reflected processes; penalisation; path decomposition; conditioning to stay negative/positive; conditioning to hit $0$ continuously; Lévy process; stable process; reflected process; penalisation; Chaumont’s -transform process; path decomposition; conditioning to stay positive/negative; conditioning to hit 0 continuously},
language = {eng},
number = {4},
pages = {1014-1032},
publisher = {Gauthier-Villars},
title = {A remarkable $\sigma $-finite measure unifying supremum penalisations for a stable Lévy process},
url = {http://eudml.org/doc/272042},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Yano, Yuko
TI - A remarkable $\sigma $-finite measure unifying supremum penalisations for a stable Lévy process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1014
EP - 1032
AB - The $\sigma $-finite measure $\mathcal {P} _{\sup }$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$-transform processes with respect to these functions are utilized for the construction of $\mathcal {P} _{\sup }$.
LA - eng
KW - Lévy processes; stable Lévy processes; reflected processes; penalisation; path decomposition; conditioning to stay negative/positive; conditioning to hit $0$ continuously; Lévy process; stable process; reflected process; penalisation; Chaumont’s -transform process; path decomposition; conditioning to stay positive/negative; conditioning to hit 0 continuously
UR - http://eudml.org/doc/272042
ER -

References

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  1. [1] 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. Zbl0414.60055MR544782
  2. [2] J. Azéma and M. Yor. Le problème de Skorokhod: compléments à “Une solution simple au problème de Skorokhod.” In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) 625–633. Lecture Notes in Math. 721. Springer, Berlin, 1979. MR544832
  3. [3] J. Bertoin. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl.47 (1993) 17–35. Zbl0786.60101MR1232850
  4. [4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. MR1406564
  5. [5] N. Bingham. Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete26 (1973) 273–296. Zbl0238.60036MR415780
  6. [6] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl.64 (1996) 39–54. Zbl0879.60072MR1419491
  7. [7] L. Chaumont. Excursion normalisée, méandre et pont pour des processus stables. Bull. Sci. Math.121 (1997) 377–403. MR1465814
  8. [8] L. Chaumont. On the law of the supremum of Lévy processes. Ann. Probab.41 (2013) 1191–1217. Zbl1277.60081MR3098676
  9. [9] L. Chaumont and R. A. Doney. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005) 948–961 (electronic); corrections in 13 (2008) 1–4 (electronic). Zbl1109.60039
  10. [10] R. A. Doney. Fluctuation Theory for Lévy Processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. Lecture Notes in Math. 1897. Springer, Berlin, 2007. 
  11. [11] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. Zbl0734.60060MR1121940
  12. [12] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin, 2006. Zbl1104.60001MR2250061
  13. [13] D. Monrad and M. L. Silverstein. Stable processes: Sample function growth at a local minimum. Z. Wahrsch. Verw. Gebiete49 (1979) 177–210. Zbl0431.60041MR543993
  14. [14] J. Najnudel and A. Nikeghbali. On some properties of a universal sigma-finite measure associated with a remarkable class of submartingales. Publ. Res. Inst. Math. Sci.47 (2011) 911–936. Zbl1266.60065MR2880381
  15. [15] J. Najnudel, B. Roynette and M. Yor. A Global View of Brownian Penalisations. MSJ Memoirs 19. Mathematical Society of Japan, Tokyo, 2009. Zbl1180.60004MR2528440
  16. [16] J. Obłój. The Skorokhod embedding problem and its offsprings. Probability Surveys1 (2004) 321–390. Zbl1189.60088MR2068476
  17. [17] J. Pitman and M. Yor. Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In Itô’s Stochastic Calculus and Probability Theory 293–310. Springer, Tokyo, 1996. MR1439532
  18. [18] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. Zbl0917.60006MR1725357
  19. [19] B. Roynette, P. Vallois and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights, I. Studia Sci. Math. Hungar.43 (2006) 171–246. Zbl1121.60027MR2229621
  20. [20] B. Roynette, P. Vallois and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II. Studia Sci. Math. Hungar.43 (2006) 295–360. Zbl1121.60004MR2253307
  21. [21] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Jpn. J. Math.1 (2006) 263–290. Zbl1160.60315MR2261065
  22. [22] B. Roynette and M. Yor. Penalising Brownian Paths. Lecture Notes in Math. 1969. Springer, Berlin, 2009. MR2504013
  23. [23] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Translated from the 1990 Japanese original, Revised by the author. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, Cambridge, 1999. MR1739520
  24. [24] M. L. Silverstein. Classification of coharmonic and coinvariant functions for Lévy processes. Ann. Probab.8 (1980) 539–575. Zbl0459.60063MR573292
  25. [25] K. Yano. Two kinds of conditionings for stable Lévy processes. In Proceedings of the 1st MSJ-SI, “Probabilistic Approach to Geometry,” 493–503. Adv. Stud. Pure Math. 57. Math. Soc. Japan, Tokyo. Zbl1200.60039MR2648275
  26. [26] K. Yano. Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. Potential Anal.32 (2010) 305–341. Zbl1188.60023MR2603019
  27. [27] K. Yano, Y. Yano and M. Yor. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan61 (2009) 757–798. Zbl1180.60008MR2552915
  28. [28] K. Yano, Y. Yano and M. Yor. Penalisation of a stable Lévy process involving its one-sided supremum. Ann. Inst. H. Poincaré Probab. Statist.46 (2010) 1042–1054. Zbl1208.60046MR2744885
  29. [29] V. M. Zolotarev. One-Dimensional Stable Distributions. Translations of Mathematical Monographs 65. Amer. Math. Soc., Providence, RI, 1986. MR854867

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