Penalisation of a stable Lévy process involving its one-sided supremum
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 1042-1054

Penalisation involving the one-sided supremum for a stable Lévy process with index α∈(0, 2] is studied. We introduce the analogue of Azéma-Yor martingales for a stable Lévy process and give the law of the overall supremum under the penalised measure.

On étudie des pénalisations d'un processus de Lévy stable d'indice α∈(0, 2] qui font intervenir son supremum unilatéral. On introduit pour un processus de Lévy stable, des martingales analogues aux martingales d'Azéma-Yor pour le mouvement brownien et son supremum; ceci permet d'obtenir la loi du supremum global relativement à la mesure pénalisée.

DOI : https://doi.org/10.1214/09-AIHP339
Classification:  60B10,  60G52,  60G44
Keywords: stable Lévy processes, reflected Lévy processes, penalisation
@article{AIHPB_2010__46_4_1042_0,
     author = {Yano, Kouji and Yano, Yuko and Yor, Marc},
     title = {Penalisation of a stable L\'evy process involving its one-sided supremum},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {1042-1054},
     doi = {10.1214/09-AIHP339},
     zbl = {1208.60046},
     mrnumber = {2744885},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_1042_0}
}
Yano, Kouji; Yano, Yuko; Yor, Marc. Penalisation of a stable Lévy process involving its one-sided supremum. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 1042-1054. doi : 10.1214/09-AIHP339. http://www.numdam.org/item/AIHPB_2010__46_4_1042_0/

[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 544782 | Zbl 0414.60055

[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 544832

[3] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0938.60005

[4] N. Bingham. Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 (1973) 273-296. | MR 415780 | Zbl 0238.60036

[5] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastics Process. Appl. 64 (1996) 39-54. | MR 1419491 | Zbl 0879.60072

[6] L. Chaumont. Excursion normalisée, méandre et pont pour des processus stables. Bull. Sci. Math. 121 (1997) 377-403. | MR 1465814 | Zbl 0882.60074

[7] 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). | Zbl 1109.60039

[8] R. A. Doney. Fluctuation theory for Lévy processes. In 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. | MR 2320889 | Zbl 1128.60036

[9] R. A. Doney. A note on the supremum of a stable process. Stochastics 80 (2008) 151-155. | MR 2402160 | Zbl 1139.60022

[10] R. A. Doney and M. S. Savov. The asymptotic behaviour of densities related to the supremum of a stable process. Preprint. | MR 2599201 | Zbl 1185.60052

[11] J. Najnudel, B. Roynette and M. Yor. A Global View of Brownian Penalisations. MSJ Memoirs 19. Mathematical Society of Japan, Tokyo, 2009. | MR 2528440 | Zbl 1180.60004

[12] A. Nikeghbali and M. Yor. Doob's maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois J. Math. 50 (2006) 791-814 (electronic). | MR 2247846 | Zbl 1101.60059

[13] L. Nguyen-Ngoc and M. Yor. Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII 42-69. Lecture Notes in Math. 1857. Springer, Berlin, 2005. | MR 2126966 | Zbl 1079.60048

[14] J. Obłój. The Skorokhod embedding problem and its offsprings. Probab. Surv. 1 (2004) 321-392. | MR 2068476 | Zbl 1189.60088

[15] J. Obłój. A complete characterization of local martingales which are functions of Brownian motion and its maximum. Bernoulli 12 (2006) 955-969. | MR 2274851 | Zbl 1130.60050

[16] J. Obłój and M. Yor. On local martingale and its supremum: Harmonic functions and beyond. In From Stochastic Calculus to Mathematical Finance 517-533. Springer, Berlin, 2006. | MR 2234288 | Zbl 1120.60045

[17] D. Ray. Stable processes with an absorbing barrier. Trans. Amer. Math. Soc. 89 (1958) 16-24. | MR 105178 | Zbl 0083.13804

[18] 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 2229621 | Zbl 1121.60027

[19] 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 2253307 | Zbl 1121.60004

[20] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Jpn. J. Math. 1 (2006) 263-290. | MR 2261065 | Zbl 1160.60315

[21] B. Roynette and M. Yor. Penalising Brownian Paths. Lecture Notes in Math. 1969. Springer, Berlin, 2009. | MR 2504013 | Zbl 1190.60002

[22] M. L. Silverstein. Classification of coharmonic and coinvariant functions for Lévy processes. Ann. Probab. 8 (1980) 539-575. | MR 573292 | Zbl 0459.60063

[23] K. Yano. Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. Potential Anal. To appear. | MR 2603019 | Zbl 1188.60023

[24] K. Yano. Two kinds of conditionings for stable Lévy processes. In Proceedings of the 1st MSJ-SI, “Probabilistic Approach to Geometry”. Adv. Stud. Pure Math. Math. Soc. Japan. | Zbl 1200.60039

[25] K. Yano, Y. Yano and M. Yor. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan 61 (2009) 757-798. | MR 2552915 | Zbl 1180.60008

[26] Y. Yano. On a remarkable σ-finite measure which unifies the supremum penalisation for a stable Lévy processes. In preparation.