On Wiener-Hopf factors for stable processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 9-19.

Nous donnons un développement en série du logarithme de l'exposant de Laplace bivarié κ des processus α-stables pour presque tous α ∈ (0, 2].

We give a series representation of the logarithm of the bivariate Laplace exponent κ of α-stable processes for almost all α ∈ (0, 2].

DOI : 10.1214/09-AIHP348
Classification : 60G51, 60E10
Mots clés : stable process, Wiener-Hopf factorization
@article{AIHPB_2011__47_1_9_0,
     author = {Graczyk, Piotr and Jakubowski, Tomasz},
     title = {On {Wiener-Hopf} factors for stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {9--19},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     doi = {10.1214/09-AIHP348},
     mrnumber = {2779394},
     zbl = {1208.60044},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/09-AIHP348/}
}
TY  - JOUR
AU  - Graczyk, Piotr
AU  - Jakubowski, Tomasz
TI  - On Wiener-Hopf factors for stable processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 9
EP  - 19
VL  - 47
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/09-AIHP348/
DO  - 10.1214/09-AIHP348
LA  - en
ID  - AIHPB_2011__47_1_9_0
ER  - 
%0 Journal Article
%A Graczyk, Piotr
%A Jakubowski, Tomasz
%T On Wiener-Hopf factors for stable processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 9-19
%V 47
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/09-AIHP348/
%R 10.1214/09-AIHP348
%G en
%F AIHPB_2011__47_1_9_0
Graczyk, Piotr; Jakubowski, Tomasz. On Wiener-Hopf factors for stable processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 9-19. doi : 10.1214/09-AIHP348. http://www.numdam.org/articles/10.1214/09-AIHP348/

[1] A. Baker. A Concise Introduction to the Theory of Numbers. Cambridge Univ. Press, Cambridge, 1984. | MR | Zbl

[2] V. Bernyk, R. C. Dalang and G. Peskir. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 (2008) 1777-1789. | MR | Zbl

[3] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

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

[5] F. Caravenna and L. Chaumont. Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 170-190. | Numdam | MR | Zbl

[6] L. Chaumont, A. E. Kyprianou and J. C. Pardo. Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 (2009) 980-1000. | MR | Zbl

[7] D. A. Darling. The maximum of sums of stable random variables. Trans. Amer. Math. Soc. 83 (1956) 164-169. | MR | Zbl

[8] R. A. Doney. On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15 (1987) 1352-1362. | MR | Zbl

[9] P. Graczyk and T. Jakubowski. On exit time of symmetric α-stable processes. Preprint, 2009.

[10] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 7th edition. Elsevier/Academic Press, Amsterdam, 2007. | MR | Zbl

[11] A. Kuznetsov. Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. J. Appl. Probab. (2009). To appear. | MR

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

[13] A. E. Kyprianou and Z. Palmowski. Fluctuations of spectrally negative Markov additive processes. In Séminaire de probabilités XLI. Lecture Notes in Math. 1934 121-135. Springer, Berlin, 2008. | MR | Zbl

[14] M. Waldschmidt. Private communication, 2009.

Cité par Sources :