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The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

  • Konstantin BorovkovEmail author
  • Zbigniew Palmowski
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 2)

Abstract

For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivariate ruin problem introduced in Avram et al. (Ann Appl Probab 18:2421–2449, 2008) in the two-dimensional case. Our solution relies on the analysis from Pan and Borovkov (Preprint. arXiv:1708.09605, 2017) for multivariate random walks and an appropriate time discretization.

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Notes

Acknowledgements

The authors are grateful to the international Mathematical Research Institute MATRIX for hosting and supporting the Mathematics of Risk program during which they obtained the result presented in this note. This work was partially supported by Polish National Science Centre Grant No. 2015/17/B/ST1/01102 (2016–2019) and the ARC Discovery grant DP150102758.

The authors are also grateful to Enkelejd Hashorva who pointed at a bug in the original version of the note.

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Dept. of Applied MathematicsWrocław University of Science and TechnologyWrocławPoland

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