Fonctions de Mittag-Leffler et processus de L\'evy stables sans saut n\'egatif

TL;DR

This paper explores the properties of Mittag-Leffler functions and their connection to Levy stable processes without negative jumps, providing new identities and explicit expressions for small deviation constants.

Contribution

It establishes a law identity between suprema of asymmetric Levy stable processes and derives explicit small deviation constants in the spectrally positive case.

Findings

Complete monotonicity of a transform of Mittag-Leffler function for a in [1,2]

Identity in law between suprema of asymmetric Levy stable processes

Explicit expression of small deviation constant for spectrally positive case

Abstract

It is noticed that a certain transform of the Mittag-Leffler function Ea is completely monotone for a in [1,2]. Using the explicit expressions of its Bernstein density, an identity in law between suprema of completely asymmetric Levy a-stable processes. In the spectrally positive case, we retrieve the exact expression of a unilateral small deviation constant which had been previously obtained by a different method by Bernyk, Dalang and Peskir.