Electronic Communications in Probability

Explicit formula for the supremum distribution of a spectrally negative stable process

Zbigniew Michna

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In this article we get simple formulas for $E\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $P(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha\,P(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative $\alpha$-stable Lévy process with $1<\alpha\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive Lévy process which follows easily from the elementary Seal's formula.


Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 10, 6 pp.

Accepted: 2 February 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G52: Stable processes 60G70: Extreme value theory; extremal processes

Lévy process distribution of the supremum of a stochastic process $\alpha$-stable Lévy process

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Michna, Zbigniew. Explicit formula for the supremum distribution of a spectrally negative stable process. Electron. Commun. Probab. 18 (2013), paper no. 10, 6 pp. doi:10.1214/ECP.v18-2236. https://projecteuclid.org/euclid.ecp/1465315549

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