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Explicit formula for the supremum distribution of a spectrally negative stable process
- Publication Year :
- 2012
-
Abstract
- In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative L\'evy 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 $\Prob(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha \Prob(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative L\'evy 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 L\'evy process which follows easily from the elementary Seals formula.
- Subjects :
- Mathematics - Probability
60G51 (Primary) 60G52, 60G70 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1206.5910
- Document Type :
- Working Paper