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Mittag-Leffler functions and stable Lévy processes without negative jumps. (Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs.) (French) Zbl 1194.33022
Summary: We remark that a certain transformation of the Mittag-Leffler function $\alpha$ is completely monotone for every $\alpha \in \left[1,2\right]$. Thanks to the exact expression of its Bernstein density function, we obtain an identity in law between one-sided exit times for completely asymmetric stable Lévy processes. In the spectrally positive case, this identity gives an expression for the density of the running supremum which is different from the one recently obtained by V. Bernyk, R. C. Dalang and G. Peskir [Ann. Probab. 36, No. 5, 1777–1789 (2008; Zbl 1185.60051)].
##### MSC:
 33E12 Mittag-Leffler functions and generalizations 60E05 General theory of probability distributions 60G52 Stable processes