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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References
Avram, F., Palmowski, Z., Pistorius, M.R.: Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann. Appl. Probab. 18, 2421–2449 (2008)
Bertoin, J., Doney, R.A.: Cramérs estimate for Lévy processes. Stat. Probab. Lett. 21, 363–365 (1994)
Borovkov, A.A.: Probability Theory, 2nd edn. Springer, London (2013)
Borovkov, A.A., Mogulskiî, A.A.: The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks. Sib. Math. J. 37, 745–782 (1996)
Borovkov, A.A., Mogulskiî, A.A.: Limit theorems in the boundary hitting problem for a multidimensional random walk. Sib. Math. J. 42, 245–270 (2001)
Collamore, J.F.: Hitting probabilities and large deviations. Ann. Probab. 24, 2065–2078 (1996)
Palmowski, Z., Pistorius, M.: Cramér asymptotics for finite time first passage probabilities of general Lévy processes. Stat. Probab. Lett. 79, 1752–1758 (2009)
Pan, Y., Borovkov, K.: The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem. Preprint. arXiv:1708.09605v2 (2017)
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.
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Borovkov, K., Palmowski, Z. (2019). The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_20
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DOI: https://doi.org/10.1007/978-3-030-04161-8_20
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