How to cite item

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

   
@article{ECP2236,
	author = {Zbigniew Michna},
	title = {Explicit formula for the supremum distribution of a spectrally negative stable process},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {18},
	year = {2013},
	keywords = {Lévy process, distribution of the supremum of a stochastic process, \$\alpha\$-stable Lévy process},
	abstract = {
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.
 
},
	pages = {no. 10, 1-6},
	issn = {1083-589X},
	doi = {10.1214/ECP.v18-2236},    
        url = {http://ecp.ejpecp.org/article/view/2236}}