A remarkable σ-finite measure unifying supremum penalisations for a stable Lévy process
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1014-1032.

On introduit la mesure σ-finie 𝒫 sup , unifiant les pénalisations selon le supremum pour un processus de Lévy stable. Dans la construction de 𝒫 sup on utilise les fonctions co-invariantes et co-harmoniques de Silverstein pour les processus de Lévy, et les processus h-transformés par rapport à ces fonctions selon l’approche de Chaumont.

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 .

DOI : 10.1214/12-AIHP497
Classification : 60G17, 60G51, 60G52, 60G44
Mots clés : Lévy processes, stable Lévy processes, reflected processes, penalisation, path decomposition, conditioning to stay negative/positive, conditioning to hit $0$ continuously
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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, Tome 49 (2013) no. 4, pp. 1014-1032. doi : 10.1214/12-AIHP497. http://archive.numdam.org/articles/10.1214/12-AIHP497/

[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. | Numdam | MR

[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. | Numdam | MR

[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. | MR

[4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR

[5] N. Bingham. Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 (1973) 273-296. | MR

[6] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR

[7] L. Chaumont. Excursion normalisée, méandre et pont pour des processus stables. Bull. Sci. Math. 121 (1997) 377-403. | MR

[8] L. Chaumont. On the law of the supremum of Lévy processes. Ann. Probab. 41 (2013) 1191-1217. | MR

[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).

[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] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. | MR

[12] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin, 2006. | MR

[13] D. Monrad and M. L. Silverstein. Stable processes: Sample function growth at a local minimum. Z. Wahrsch. Verw. Gebiete 49 (1979) 177-210. | MR

[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. | MR

[15] J. Najnudel, B. Roynette and M. Yor. A Global View of Brownian Penalisations. MSJ Memoirs 19. Mathematical Society of Japan, Tokyo, 2009. | MR

[16] J. Obłój. The Skorokhod embedding problem and its offsprings. Probability Surveys 1 (2004) 321-390. | MR

[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. | MR

[18] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | MR

[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. | MR

[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. | MR

[21] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Jpn. J. Math. 1 (2006) 263-290. | MR

[22] B. Roynette and M. Yor. Penalising Brownian Paths. Lecture Notes in Math. 1969. Springer, Berlin, 2009. | MR

[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. | MR

[24] M. L. Silverstein. Classification of coharmonic and coinvariant functions for Lévy processes. Ann. Probab. 8 (1980) 539-575. | MR

[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. | MR

[26] K. Yano. Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. Potential Anal. 32 (2010) 305-341. | MR

[27] K. Yano, Y. Yano and M. Yor. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan 61 (2009) 757-798. | MR

[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. | Numdam | MR

[29] V. M. Zolotarev. One-Dimensional Stable Distributions. Translations of Mathematical Monographs 65. Amer. Math. Soc., Providence, RI, 1986. | MR

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