On exponential functionals of processes with independent increments

Abstract : In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive necessary and sufficient conditions for the existence of the Laplace exponent of $I_t$, and also the sufficient conditions of finiteness of the Mellin transform ${\bf E}(I_t^{\alpha})$ with $\alpha\in \mathbb{R}$. We give a recurrent integral equations for this Mellin transform. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponentials of Levy processes, which hold under less restrictive conditions then in \cite{BY}. In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we precise the exact number of finite moments of $I_{\infty}$.
Type de document :
Pré-publication, Document de travail
26 pages, no figures. 2016
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Contributeur : Lioudmila Vostrikova <>
Soumis le : vendredi 14 juillet 2017 - 21:09:56
Dernière modification le : mercredi 21 novembre 2018 - 11:55:52

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  • HAL Id : hal-01388080, version 2
  • ARXIV : 1610.08732

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Paavo Salminen, L Vostrikova. On exponential functionals of processes with independent increments. 26 pages, no figures. 2016. 〈hal-01388080v2〉

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