no. 5
On suprema of Lévy processes and application in risk theory
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 977-986.

Soit Y un processus de Lévy réel quelconque et C un subordinateur indépendant de Y. On considère les temps en lesquels le processus X ^=C-Y atteint un nouveau maximum par un saut de C. Nous donnons une condition nécessaire et suffisante pour que l’ensemble de ces temps soit discret. Lorsque tel est le cas et que le processus X ^ dérive vers -∞, nous décomposons son maximum absolu en cette suite de temps. Nous déduisons alors de cette décomposition une formule du type Pollaczek-Hinchin pour la loi du maximum absolu de X ^.

Let X ^=C-Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X ^ is reached by a jump of the subordinator C. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and X ^ drifts to -∞, we decompose the absolute supremum of X ^ at these times, and derive a Pollaczek-Hinchin-type formula for the distribution function of the supremum.

DOI : 10.1214/07-AIHP142
Classification : 60G51, 60G17, 60J75, 91B30
Mots clés : Lévy process, subordinator, fluctuation theory, extrema, risk theory 
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Song, Renming; Vondraček, Zoran. On suprema of Lévy processes and application in risk theory. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 977-986. doi : 10.1214/07-AIHP142. http://archive.numdam.org/articles/10.1214/07-AIHP142/

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

[2] J. Bertoin. Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121 (1997) 345-354. | MR | Zbl

[3] J. Bertoin. Regenerative embedding of Markov sets. Probab. Theory Related Fields 108 (1997) 559-571. | MR | Zbl

[4] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14 (2004) 1378-1397. | MR | Zbl

[5] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities for competing claim processes. J. Appl. Probab. 41 (2004) 679-690. | MR | Zbl

[6] C. Klüppelberg, A. Kyprianou and R. Maller. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004) 1766-1801. | MR | Zbl

[7] C. Klüppelberg and A. Kyprianou. On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006) 594-598. | MR | Zbl

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

[9] V. Vigon. Votre Lévy rampe-t-il? J. London Math. Soc. (2) 65 (2002) 243-256. | MR | Zbl

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