Open Access
2015 The distribution of the supremum for spectrally asymmetric Lévy processes
Zbigniew Michna, Zbigniew Palmowski, Martijn Pistorius
Author Affiliations +
Electron. Commun. Probab. 20: 1-10 (2015). DOI: 10.1214/ECP.v20-2999


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.


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Zbigniew Michna. Zbigniew Palmowski. Martijn Pistorius. "The distribution of the supremum for spectrally asymmetric Lévy processes." Electron. Commun. Probab. 20 1 - 10, 2015.


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

zbMATH: 1321.60097
MathSciNet: MR3327863
Digital Object Identifier: 10.1214/ECP.v20-2999

Primary: 60G51
Secondary: 60G70

Keywords: distribution of the supremum of a stochastic process , Lévy process , spectrally asymmetric Lévy process

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