Electronic Communications in Probability

The distribution of the supremum for spectrally asymmetric Lévy processes

Zbigniew Michna, Zbigniew Palmowski, and Martijn Pistorius

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In this article we derive formulas for the probability $\mathbb{P}(\sup_{t\leq T} X(t)>u)$, $T>0$ and $\mathbb{P}(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.

Article information

Electron. Commun. Probab. Volume 20 (2015), paper no. 24, 10 pp.

Accepted: 13 March 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G70: Extreme value theory; extremal processes

Lévy process distribution of the supremum of a stochastic process spectrally asymmetric Lévy process

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Michna, Zbigniew; Palmowski, Zbigniew; Pistorius, Martijn. The distribution of the supremum for spectrally asymmetric Lévy processes. Electron. Commun. Probab. 20 (2015), paper no. 24, 10 pp. doi:10.1214/ECP.v20-2999. https://projecteuclid.org/euclid.ecp/1465320951

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  • Asmussen, S.; Albrecher, H. Ruin probabilities. Second edition. Advanced Series on Statistical Science & Applied Probability, 14. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. xviii+602 pp. ISBN: 978-981-4282-52-9; 981-4282-52-9
  • Bernyk, Violetta; Dalang, Robert C.; Peskir, Goran. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 (2008), no. 5, 1777–1789.
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
  • Bertoin, J.; Doney, R. A.; Maller, R. A. Passage of Lévy processes across power law boundaries at small times. Ann. Probab. 36 (2008), no. 1, 160–197.
  • Borovkov, K.; Burq, Z. Kendall's identity for the first crossing time revisited. Electron. Comm. Probab. 6 (2001), 91–94.
  • Chung, K. L.; Fuchs, W. H. J. On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 1951, (1951). no. 6, 12 pp.
  • Furrer, H.: Risk theory and heavy-tailed Lévy processes. Ph.D. Thesis ETH Zürich, 1997.
  • Furrer, H. Risk processes perturbed by $\alpha$-stable Lévy motion. Scand. Actuar. J. 1998, no. 1, 59–74.
  • Harrison, J. Michael. The supremum distribution of a Lévy process with no negative jumps. Advances in Appl. Probability 9 (1977), no. 2, 417–422.
  • Hubalek, Friedrich; Kuznetsov, Alexey. A convergent series representation for the density of the supremum of a stable process. Electron. Commun. Probab. 16 (2011), 84–95.
  • Kendall, David G. Some problems in theory of dams. J. Roy. Statist. Soc. Ser. B. 19 (1957), 207–212; discussion 212–233.
  • Kuznetsov, Alexey. Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 (2010), no. 5, 1801–1830.
  • KwaÅ›nicki, Mateusz; Małecki, Jacek; Ryznar, Michał. Suprema of Lévy processes. Ann. Probab. 41 (2013), no. 3B, 2047–2065.
  • Michna, Zbigniew. Formula for the supremum distribution of a spectrally positive $\alpha$-stable Lévy process. Statist. Probab. Lett. 81 (2011), no. 2, 231–235.
  • Michna, Zbigniew. Explicit formula for the supremum distribution of a spectrally negative stable process. Electron. Commun. Probab. 18 (2013), no. 10, 6 pp.
  • Nolan, John P. Numerical calculation of stable densities and distribution functions. Heavy tails and highly volatile phenomena. Comm. Statist. Stochastic Models 13 (1997), no. 4, 759–774.
  • Prabhu, N. U. On the ruin problem of collective risk theory. Ann. Math. Statist. 32 1961 757–764.
  • Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
  • Seal, Hilary L. The numerical calculation of $U(w,\,t)$, the probability of non-ruin in an interval $(O,\,t)$. Scand. Actuar. J. 1974, 121–139.
  • Simon, Thomas. Hitting densities for spectrally positive stable processes. Stochastics 83 (2011), no. 2, 203–214.
  • Takacs, Lajos. On the distribution of the supremum for stochastic processes with interchangeable increments. Trans. Amer. Math. Soc. 119 1965 367–379.
  • Zolotarev, V. M. The moment of first passage of a level and the behaviour at infinity of a class of processes with independent increments. (Russian) Teor. Verojatnost. i Primenen. 9 1964 724–733.