On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes

Abstract

pp. 33-45 R,ESUME.-On calcule les moments entiers positifs de certains processus de Markov auto-similaires évalués en un temps fixe, ainsi que les moments entiers negatifs des fonctionnelles exponentielles de certains processus de Levy.Lorsque le processus de Levy sous-jacent n'a pas de saut positif, ces moments entiers determinent la loi de la variable et conduisent a d'interessantes identites en distribution.Le cas du processus de Poisson donne un nouvel exemple simple qui montre que la loi log-normale n'est pas determinee par ses moments entiers.ABSTRACT.-We compute the positive entire moments of certain self- similar Markov processes evaluated at fixed time, and the negative entire moments of the exponential functional I of certain Levy processes.When the Levy process has no positive jumps, this determines the aforemen- tioned distributions and yields several interesting identities in law.The case of the Poisson process yields yet another simple example showing that the log-normal distribution is moment-indeterminate.

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