no. 4
Penalisation of a stable Lévy process involving its one-sided supremum
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 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 : 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},
     pages = {1042--1054},
     year = {2010},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     doi = {10.1214/09-AIHP339},
     mrnumber = {2744885},
     zbl = {1208.60046},
     language = {en},
     url = {https://www.numdam.org/articles/10.1214/09-AIHP339/}
}
TY  - JOUR
AU  - Yano, Kouji
AU  - Yano, Yuko
AU  - Yor, Marc
TI  - Penalisation of a stable Lévy process involving its one-sided supremum
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2010
SP  - 1042
EP  - 1054
VL  - 46
IS  - 4
PB  - Gauthier-Villars
UR  - https://www.numdam.org/articles/10.1214/09-AIHP339/
DO  - 10.1214/09-AIHP339
LA  - en
ID  - AIHPB_2010__46_4_1042_0
ER  - 
%0 Journal Article
%A Yano, Kouji
%A Yano, Yuko
%A Yor, Marc
%T Penalisation of a stable Lévy process involving its one-sided supremum
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 1042-1054
%V 46
%N 4
%I Gauthier-Villars
%U https://www.numdam.org/articles/10.1214/09-AIHP339/
%R 10.1214/09-AIHP339
%G en
%F 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, Tome 46 (2010) no. 4, pp. 1042-1054. doi: 10.1214/09-AIHP339

[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. | Zbl | MR | Numdam

[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. | MR | Numdam

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

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

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

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

[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

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

[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

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

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

Cité par Sources :