On suprema of Lévy processes and application in risk theory

Renming Song; Zoran Vondraček

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 5, page 977-986
  • ISSN: 0246-0203

Abstract

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Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ 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 X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.

How to cite

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Song, Renming, and Vondraček, Zoran. "On suprema of Lévy processes and application in risk theory." Annales de l'I.H.P. Probabilités et statistiques 44.5 (2008): 977-986. <http://eudml.org/doc/78000>.

@article{Song2008,
abstract = {Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ 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 X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.},
author = {Song, Renming, Vondraček, Zoran},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy process; subordinator; fluctuation theory; extrema; risk theory},
language = {eng},
number = {5},
pages = {977-986},
publisher = {Gauthier-Villars},
title = {On suprema of Lévy processes and application in risk theory},
url = {http://eudml.org/doc/78000},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Song, Renming
AU - Vondraček, Zoran
TI - On suprema of Lévy processes and application in risk theory
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 5
SP - 977
EP - 986
AB - Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ 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 X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.
LA - eng
KW - Lévy process; subordinator; fluctuation theory; extrema; risk theory
UR - http://eudml.org/doc/78000
ER -

References

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  1. [1] J. Bertoin. Lévy Processes. Cambridge Univ. Press, 1996. Zbl0861.60003MR1406564
  2. [2] J. Bertoin. Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121 (1997) 345–354. Zbl0883.60069MR1465812
  3. [3] J. Bertoin. Regenerative embedding of Markov sets. Probab. Theory Related Fields 108 (1997) 559–571. Zbl0895.60011MR1465642
  4. [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. Zbl1061.60075MR2071427
  5. [5] M. Huzak, M. Perman, H. Šikić and Z. Vondraček. Ruin probabilities for competing claim processes. J. Appl. Probab. 41 (2004) 679–690. Zbl1065.60100MR2074816
  6. [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. Zbl1066.60049MR2099651
  7. [7] C. Klüppelberg and A. Kyprianou. On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006) 594–598. Zbl1118.60071MR2248586
  8. [8] A. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. Zbl1104.60001MR2250061
  9. [9] V. Vigon. Votre Lévy rampe-t-il? J. London Math. Soc. (2) 65 (2002) 243–256. Zbl1016.60054MR1875147

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