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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320951

Digital Object Identifier
doi:10.1214/ECP.v20-2999

Mathematical Reviews number (MathSciNet)
MR3327863

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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